Print the puzzles or type the solution on this excel file: 12 factors 829-834
Before I write a blog post, I look to see how the post number is expressed in different bases. Today I noticed that 831 is 30333 in BASE 4. I was intrigued by all those 3’s because I knew that 831 = 3 × 277. It seems logical that 277 would be 10111 in BASE 4, it turns out that it is! I looked at 831 in all the bases up to BASE 36. Did any others have only multiples of 3 as its digits? Yes, a few did, so I’ve made a chart of 277 and 831 in those five bases to make comparing them easy. I also used only base 10 numbers and not letters of the alphabet to represent the digits in the other bases. As you look at this chart, remember 3 × 277 = 831.
Why are those the ONLY bases for which 3 times the digits of 277 equals the digits for 831? Because in every other base, at least one of the digits times 3 will be greater than or equal to the base and some complicated carrying will have to take place to determine the digits for 831 in that base.
For example, 277 is palindrome 1 11 1 in BASE 12. Obviously 3 times 1 11 1 is 3 33 3. Since 33 is bigger than 12, we somehow end up with non-palindrome 59 3 in BASE 12 for 831. This is how that somehow happened: 33÷12 = 2R9. The 9 becomes the middle digit while the 2 is added to the original 3 to make the first digit, 5.
Here’s a little more about the number 831:
Because 277 is one of its factors, 831 is the hypotenuse of a Pythagorean triple: 345-756-831, which is 3 times primitive (115-252-277).
831 is a composite number.
Prime factorization: 831 = 3 × 277
The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 831 has exactly 4 factors.
Factors of 831: 1, 3, 277, 831
Factor pairs: 831 = 1 × 831 or 3 × 277
831 has no square factors that allow its square root to be simplified. √831 ≈ 8270706
Print the puzzles or type the solution in this excel file: 12 factors 829-834
Whether you write today’s date as 8-15-17 or 15-8-17, today is (Primitive) Pythagorean Triple Day. 829 is also in a couple of primitive Pythagorean triples:
829 is the last prime number in the fourth prime decade, (821, 823, 827, 829). Later this year we can celebrate the FIRST prime decade on 11-13-17 at 1900 hours.
Sara Van Der Werf wrote a post that includes a poster of these and other mathematical holidays that can be celebrated in the 2017-18 school year. She will also keep us posted on ways to celebrate these holidays as each one approaches. The poster is a word document that can be edited if need be.
829 is also the sum of three consecutive prime numbers: 271 + 277 + 281 = 829
829 is the 24th centered triangular number. I challenged myself to make a graphic that would show the significance of this fun number fact.
Now you can see why it is called a CENTERED triangular number. If you count these concentric triangles including the tiny, barely visible one in the very center, you will see that there are 24 of them. Can you also see that 829 is equal to the sum of the 22nd, 23rd, and 24th triangular numbers?
What else is true because 829 is the 24th centered triangular number?
(22×23 + 23×24 + 24×25)/2 = 829.
3(23×24)/2 + 1 = 829
Here is 829’s factoring information:
829 is a prime number.
Prime factorization: 829 is prime.
The exponent of prime number 829 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 829 has exactly 2 factors.
Factors of 829: 1, 829
Factor pairs: 829 = 1 × 829
829 has no square factors that allow its square root to be simplified. √829 ≈ 28.79236
How do we know that 829 is a prime number? If 829 were not a prime number, then it would be divisible by at least one prime number less than or equal to √829 ≈ 28.8. Since 829 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 829 is a prime number.
Here’s another way we know that 829 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 27² + 10² = 829 with 27 and 10 having no common prime factors, 829 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √829 ≈ 28.8. Since 829 is not divisible by 5, 13, or 17, we know that 829 is a prime number.
828 is the sum of consecutive prime numbers 409 and 419.
828 has a lot of factors so I decided to use it in my examples of synthetic division. What are the factors of 828?
828 is a composite number.
Prime factorization: 828 = 2 × 2 × 3 × 3 × 23, which can be written 828 = 2² × 3² × 23
The exponents in the prime factorization are 2, 2 and 1. Adding one to each and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 × 3 × 2 = 18. Therefore 828 has exactly 18 factors.
Taking the factor pair with the largest square number factor, we get √828 = (√36)(√23) = 6√23 ≈ 28.774989.
Synthetic division is taught in many schools in the United States, but in other places in the world it typically is not taught at all. Although I prefer the area model for dividing polynomials, I still like synthetic division. I disagree with those few people who describe it as a mostly useless trick that isn’t worth learning. Yes, its usefulness is limited, but when it can be used, it can be absolutely wonderful. Personally, for many years I always use synthetic division when dividing polynomials by (x-a) or (x+a) where a is any whole number. (If a is a fraction, synthetic division can still be done, but it might not be much fun.)
What are some of the advantages of using synthetic division?
If you had a polynomial where x is raised to several different powers, such as x⁹ + x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x – 8, you would only have to write 1 1 1 1 1 1 1 1 1 -8 to perform the algorithm. That could prevent writer’s cramp if the polynomial is quite long. Ha ha. Seriously, less writing often means fewer chances for mistakes.
Instead of needing 9×2 lines to do long division for the problem, only three total lines are needed. That saves paper.
Using a instead of (x-a) or -a instead of (x+a) in the algorithm means we use addition instead of subtraction to find the quotient. Most people make fewer mistakes adding numbers than they do subtracting. Fewer mistakes means less frustration and less erasing.
Before we can do synthetic division we need to write some polynomials. Since this is my 828th post, I will write some polynomials based on the following chart, and they will be very special polynomials!
The numbers in bold print end in a zero because the corresponding base number is a factor of 828. For base 11 or greater, sometimes a digit is represented by a letter of the alphabet. The key to translating those letters to the corresponding number in base 10 is A = 10, B = 11, C = 12, D = 13, E = 14, F = 15, G = 16, H = 17, I = 18, J = 19, K = 20, L = 21, M = 22. This chart goes to BASE 28 because √828 ≈ 28.77.
We can write a special polynomial using the digits given from a desired base. The last digit for these special polynomials will be replaced with -828, but as you will see, that original last digit will not be forgotten.
Because 828 is 30330 in BASE 4, let’s use that information as our first example:
The digits 30330 make the polynomial 3x⁴ + 0x³ + 3x² + 3x -828.
The digits 3 0 3 3 -828 will be used as the coefficients in our synthetic division algorithm.
BASE 4 will be seen in the divisor (x – 4) and as “4” in the algorithm.
Now watch as this gif uses synthetic division to find the quotient.
The remainder is zero, and the last digit of 30330 is zero. From the remainder theorem we also know that 3(4⁴) + 3(4²) + 3(4) -828 = 0.
It turns out we can know what the remainder is for each of these special polynomials BEFORE we do any dividing! The remainder will be the last digit times negative one. That does not usually happen when we use synthetic division on a polynomial, but it will always happen on these special polynomials!
Here are a four more examples of writing one of these special polynomials and dividing it using synthetic division. Try writing the rest of the problems using some of the other bases and doing the division yourself, too.
Now for today’s Find the Factors puzzle:
Print the puzzles or type the solution on this excel file: 10-factors-822-828
It’s not an easy puzzle! If you get stumped, here is the logic I used to solve it:
Clue 27 will use a 3, so clue 9 cannot be 3×3. Thus, clues 9 and 18 will put 9 in the first column and 1 and 2 in the top row.
Can both 40’s be 4×10? No, because that would use both 10’s, and make the 8 and the 18 use both 2’s. That would mean that clue 10 could not be 10×1 or 2×5.
So 56 and one of the 40’s will use both 8’s. That means 24 has to use 4 and 6. Thus 24 and 42 will use both 6’s, so 30 will be 10×3.
We know one of the 40’s is 4×10, but we don’t know which one. Nevertheless, we know that its 4 will be in the first column because its 10 cannot be. Since 24 must use 4 and 6, its 4 must be in the top row above the 24.
This table shows the rest of the logic I used:
That was pretty complicated, so here’s where all the factors go, too. 🙂
Numbers ending in 00, 25, 50, and 75 can be divided evenly by 25. How much is 825 divided by 25? That quotient is the same as the answer to “how many quarters are in $8.25?” (A quarter is ¼ of a dollar and is written .25 or 25¢.)
You probably could visualize the answer in your head even if I hadn’t included a picture! That’s why I often ask kids the how-many-quarters question when they are stumped dividing something by 25 . It seems that kids are always able to give the quotient after that question. Notice that “8.25 ÷ .25 =” and “8 ¼ ÷ ¼ =” have the same answer, too. You can also ask that how-many-quarters question to find the answer when something is divided by .25 or ¼.
It would almost be as easy to divide $8.26 or $8.39 by 25. The quotient would be the same as the problem above but with some loose change becoming the remainder. Using money for division problems could even help kids better understand dividends, divisors, quotients, and remainders.
Here’s an example of a how-many-quarters type question that will help you divide by 75, .75 or ¾.
We can count and see that there are 11 sets of 3 quarters in $8.25. That means that $8.25 ÷ .75 is 11. It also means that 8¼ ÷ ¾ = 11.
Dividing by fractions can be a very abstract concept for students, and they may ask questions like, “What does 8¼ ÷ 1¼ even mean?” Again quarters come to the rescue! 5 quarters can be so much more friendly than 1¼ is. Shorthand for 5 quarters is the fraction, 5/4. Since they have the same denominators, dividing 8¼ by 1¼ is the same as dividing 33 by 5:
Kids think money is more fun than math, but money is just a subset of mathematics which is full of lots of other fun topics, too. Here are a couple of ways other educators have used money to teach a math topic.
Jen of Beyond Tradit’l Math shared her way to teach subtracting decimals using money. Her method will surely captivate any child who tries it and even make regrouping fun:
Robert Kaplinsky uses several very short videos to keep students engaged without them actually touching any money. Check out the replies, too. Paula Beardell Krieg’s excellent $1.00 art project is there, and that would be fun for anyone 2nd grade or older to do:
Clearly 825 has to be divisible by both 5 AND 3 in order for (821, 823, 827, 829) to be the fourth prime decade, which it is.
The last digit of 825 is 5, so it is divisible by 5.
8 + 2 + 5 = 15, a multiple of 3, so 825 is divisible by 3.
8 – 2 + 5 = 11, so 825 is divisible by 11.
All numbers ending in 00, 25, 50, or 75 can have their square roots simplified. If you were trying to simplify √825, you could visualize quarters in your mind to easily divide 825 by 25. Then √825 = (√25)(√33) = 5√33
Here is 825’s factoring information:
825 is a composite number.
Prime factorization: 825 = 3 × 5 × 5 × 11, which can be written 825 = 3 × 5² × 11
The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 825 has exactly 12 factors.
Perhaps you know that of the three numbers in EVERY Pythagorean triple at least one of them will be divisible by 3, at least one of them will be divisible by 4, and at least one of them will be divisible by 5. That’s obvious for triple 3-4-5.
Here’s another example: Pythagorean triple 11-60-61. Two of those numbers are prime numbers, yet 60 is divisible by 3, 4, AND 5.
If the Pythagorean triple is a primitive, something else amazing happens:
That’s amazing all by itself, but that’s only part of the picture. Let’s look at the complete picture specifically using the triple with hypotenuse 821:
25² + 14² = 821
429² + 700² = 821²
429–700–821 can be calculated from 25² – 14², 2(25)(14), 25² + 14²
821 + 700 = 1521
√1521 = 39 which is 25 + 14
429 ÷ 39 = 11
WHY does this happen to primitive Pythagorean triples?
It happened for 429–700–821 because 700+ 821 = 25² + 14² + 2(25)(14) = (25 + 14)².
After all (a + b)² = a² +2ab + b² is always true.
And it can be rearranged: (a + b)² = a² + b² +2ab.
You can use a similar proof whenever the element of the Primitive triple that is divisible by 4 can be expressed as 2ab.
************
But it is a different, and perhaps simpler, story for many triples such as9–40–41 which was calculated using (9)(1), (9² – 1²)/2, (9² + 1²)/2.
In that case 40 + 41 = (9² – 1²)/2 + (9² + 1²)/2 = (9² – 1² + 9² + 1²)/2 = 9².
You can use a similar proof whenever the element of the Primitive triple that is divisible by 4 can be expressed as (a² – b²)/2.
************
Go ahead, take ANY primitive Pythagorean triple. Add the leg that is divisible by 4 to its hypotenuse. You will get a perfect square. Here’s a few more examples:
Similarly primitive triples that have been multiplied by a square number will also produce a perfect square, but you’ll have to be careful which leg you add to the hypotenuse if you multiplied by a square number that is a multiple of 4.
For example, 9(3-4-5) = 27-36-45 and 36 + 45 = 81, a square number.
But 4(3-4-5) = 12-16-20 so each number is divisible by 4. Note that 16 + 20 = 36, a square number, but 12 + 20 does not.
Here’s a few other essential facts about the number 821:
All of the odd numbers between 820 and 830, except 825, are prime numbers. Thus 821, 823, 827, 829 is the fourth prime decade.
821 is a prime number.
Prime factorization: 821 is prime.
The exponent of prime number 821 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 821 has exactly 2 factors.
Factors of 821: 1, 821
Factor pairs: 821 = 1 x 821
821 has no square factors that allow its square root to be simplified. √821 ≈ 28.65309756.
How do we know that 821 is a prime number? If 821 were not a prime number, then it would be divisible by at least one prime number less than or equal to √821 ≈ 28.7. Since 821 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 821 is a prime number.
Here’s another way we know that 821 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 25² + 14² = 821 with 25 and 14 having no common prime factors, 821 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √821 ≈ 28.7. Since 821 is not divisible by 5, 13, or 17, we know that 821 is a prime number.
Let’s begin with today’s puzzle. Afterwards I’ll tell you a little about the number 820 and why I decided to make a list of all the numbers up to 820 with exactly 12 factors.
Print the puzzles or type the solution on this excel file: 12 factors 815-820
Now here’s some information about the number 820:
820 is a composite number.
Prime factorization: 820 = 2 × 2 × 5 × 41, which can be written 820 = 2² × 5 × 41
The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 820 has exactly 12 factors.
Taking the factor pair with the largest square number factor, we get √820 = (√4)(√205) = 2√205 ≈ 28.635642
820 is the sum of two squares two ways:
28² + 6² = 820
26² + 12² = 820
Because 5 and 41 are two of its prime factors, 820 is the hypotenuse of FOUR Pythagorean triples:
180-800-820 which is 20 times 9-40-41
336-748-820 which is 4 times 84-187-205, but it could also be calculated from 2(28)(6), 28² – 6², 28² + 6²
492-656-820 which is 164 times 3-4-5
532-624-820 which is 4 times 133-156-205, but it could also be calculated from 26² – 12², 2(26)(12),26² + 12²
Since 820 = (40×41)/2, we know that 820 is the 40th triangular number, thus
1 + 2 + 3 + . . . + 38 + 39 + 40 = 820
820 is also a palindrome in four other bases:
1010101 BASE 3 because 3⁶+3⁴+3²+3⁰=820
1111 BASE 9 because 9³+9²+9¹+9⁰=820
868 BASE 11 because 8(11²)+6(11¹)+8(11⁰)=820
1I1 BASE 21 (I is 18 base 10) because 21²+18(21¹)+21⁰=820
Below is a chart of the numbers up to 820 with exactly 12 factors. Notice that two sets of consecutive numbers, (735, 736) and (819, 820), are on the list. Look at their prime factorizations:
735=3×5×7², 736=2⁵×23
819=3²×7×13, 820=2²×5×41
Those prime factorizations mean that while 735 and 736 are the smallest consecutive numbers with exactly 12 factors, 819 and 820 are the smallest consecutive numbers whose prime factorizations consist of one prime number squared and exactly two other primes. Thanks to OEIS.org for alerting me to that fact. Here’s something interesting about the chart: of the 77 numbers listed, only six are odd numbers.
So, how did I know what numbers to put on the list?
In order to determine how many numbers up to 820 have exactly 12 factors, we must first factor 12. We know that 12=12, 6×2, 4×3, and 3×2×2.
Next we subtract 1 from each of those factors to determine the exponents we need to use:
12 gives us 12-1=11. For prime number a, when is a¹¹ not larger than 820? Never, because 2¹¹>820.
6×2 gives us 6-1=5 and 2-1=1. For prime numbers a and b, with a≠b, when is a⁵b¹ less than or equal to 820? These nine times:
2⁵×3=96, 2⁵×5=160, 2⁵×7=224, 2⁵×11=352,
2⁵×13=416, 2⁵×17=544, 2⁵×19=608, 2⁵×23=736
3⁵×2=486
4×3 gives us 4-1=3 and 3-1=2. For prime numbers a and b, with a≠b, when is a³b² not larger than 820? These six times:
2³×3²=72, 2³×5²=200, 2³×7²=392
3³×2²=108, 3³×5²=675
5³×2²=500
3×2×2 gives us 3-1=2, 2-1=1, and 2-1=1. For distinct prime numbers a, b, and c, when is a²bc not larger than 820? 52 times. Here’s the breakdown: It happens 35 times when 2²=4 is the square number:
And it happens another 27 times when a prime number other than 2 is squared:
That’s a lot of numbers with exactly 12 factors! After I sorted all the numbers that I found in numerical order, I was able to make that chart of numbers up to 820 with exactly 12 factors, and yes 819 and 820 are the smallest two consecutive numbers whose prime factorization consists exactly of one prime number squared and two other prime numbers.
814 is even, so it is divisible by 2. What else is it divisible by? Let’s apply 11’s divisibility rule:
8 – 1 + 4 = 11, so 814 can also be evenly divided by 11. (If the sum of every other digit minus the sum of the missed digits is divisible by 11, then the number is divisible by 11.)
This Level 6 Puzzle is a doozie, but I assure you that it CAN BE be solved using logic without guessing and checking!
Print the puzzles or type the solution on this excel file: 10-factors 807-814
Now here’s a little bit more about the number 814:
814 is a composite number.
Prime factorization: 814 = 2 x 11 x 37
The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 814 has exactly 8 factors.
Factors of 814: 1, 2, 11, 22, 37, 74, 407, 814
Factor pairs: 814 = 1 x 814, 2 x 407, 11 x 74, or 22 x 37
814 has no square factors that allow its square root to be simplified. √814 ≈ 28.530685
Because 37 is one of its factors, 814 is the hypotenuse of a Pythagorean triple:
264-770-814 which is 22 times (12-35-37)
814 is also repddigit MM in BASE 36 (M is 22 base 10) because 22(36) + 22(1) = 22(36 + 1) = 22 × 37 = 814
Here are some initial logical notes to solve Find the Factors Puzzle #814:
10, 8, 4, and 6 must use all the 1’s and 2’s, so clue 4 cannot use both 2’s and must use 1 & 4.
The 8’s must be used by 72 and one of the 40’s. That 40 will also use a 5.
63 & 72 will use both 9’s so 18 must use a 3, as does 21.
That means 30 can’t be 3 x 10 and must use a 5.
Thus a 40 and 30 will use both 5’s so clue 10 can’t use a 5 and must use 1 & 10
In case there is still any confusion, here’s where the factors went:
Here’s a puzzle for you. How many total triangles are there in the figure below?
Okay, if you guessed 812 because this is my 812th post, you would be right. But what if you were asked that question in some real world situation where accessing the internet to get the answer isn’t permitted. How would you know the answer then?
The total number of triangles = ⌊n(n+2)(2n+1)/8⌋ where the brackets mean round decimals DOWN to the closest integer.
Here n = 14, so the number of triangles is 14×16×29/8 = 812. Rounding down wasn’t necessary since the product of two consecutive even numbers is always divisible by 8.
Still, you probably wouldn’t remember that formula unless you had seen it VERY recently or you have a photographic memory.
You could actually COUNT all the triangles. In the post about 658 total triangles, I noted that the 13 rows of small triangles formed a total of 169 of the smallest triangles, but I suggested that it would be easier to ignore that nice square number and instead count the number of triangles pointing UP separately from the number pointing DOWN. You will add up a lot of triangular numbers as you sum up the number of them pointing up and again as you sum up the number pointing down.
Making a chart of the number of triangles pointing UP would be easy. It’s just a list of triangular numbers in order. However, the chart for the ones pointing down might be confusing because you don’t use all of the triangular numbers, and the ones you use will be different for an even number of rows than for an odd number of rows. For example,
the pointing DOWN portion of the chart for 14 rows of triangles below uses these seven triangular numbers: 1, 6, 15, 28, 45, 66, and 91,
while the pointing DOWN chart for 13 rows of triangles uses six different triangular numbers: 3, 10, 21, 36, 55, and 78.
Here is a chart listing the number of triangles of any size that are contained in a triangular figure made with 14 rows of small triangles. Interesting note: Because 14 + 2 = 16, a multiple of 8, the total number of triangles in this case will be divisible by 14, the number of line segments on each side.
Making such a chart works. However, remembering which triangular numbers to use and how many you should use, especially when counting the odd number of rows of triangles pointing DOWN, might be difficult.
Today I was thinking about triangular numbers and their relationship to square numbers:
The sum of two consecutive triangular numbers always makes a square number. On the chart above, I’ve paired up certain consecutive triangular numbers by coloring them the same color. The sizes of the triangles being paired together by like colors are not the same size (with one exception), but their sums are nevertheless square numbers every time.
Thus, we also can come up with a single list of numbers to add to arrive at the total number of triangles as illustrated below. This has made me changed my mind about using those square numbers to help count! Notice how every other number on each list below, including the last number to be added, is a square number.
Of the methods discussed in this post, this one that includes square numbers might be the easiest one to remember. Here are the steps I used to find the total number of triangles:
Count the rows and determine if that number is even or odd.
Write the triangular numbers in order until the amount of numbers written equals the number of rows.
If the number of rows is even, replace the 2nd, 4th, 6th, etc. entries with corresponding square numbers: 4, 16, 36, etc.
If the number of rows is odd, replace the 1st, 3rd, 5th, etc entries with corresponding square numbers: 1, 9, 25, etc.
Add all the numbers remaining in the list. The sum will be the total number of triangles for that many rows of small triangles.
Okay, that puzzle was rather difficult and took a while to explain. You may find this Level 4 puzzle easier:
Print the puzzles or type the solution on this excel file: 10-factors 807-814
Here’s a little more about the number 812:
812 is a composite number.
Prime factorization: 812 = 2 x 2 x 7 x 29, which can be written 812 = 2² x 7 x 29
The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 812 has exactly 12 factors.
My youngest grandchild is 7 months old and is just beginning to crawl. He stops crawling periodically to wipe drool off his own face. It’s pretty funny.
Here’s today’s puzzle:
Print the puzzles or type the solution on this excel file: 10-factors 807-814
811 is a prime number.
Prime factorization: 811 is prime.
The exponent of prime number 811 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 811 has exactly 2 factors.
Factors of 811: 1, 811
Factor pairs: 811 = 1 x 811
811 has no square factors that allow its square root to be simplified. √811 ≈ 28.4780617
How do we know that 811 is a prime number? If 811 were not a prime number, then it would be divisible by at least one prime number less than or equal to √811 ≈ 28.5. Since 811 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 811 is a prime number.
811 is the second half of a twin prime, and it is also the sum of five consecutive prime numbers:
151 + 157 + 163 + 167 + 173 = 811.
OEIS.org informs us that the smallest prime factor of 24!+1 is 811. Wow! Really?
If you type 24!+1 into your computer calculator you get 620,448,401,733,239,439,360,001.
If you divide that number by 811, you get 765,041,185,860,961,084,291.
If you type that into Number Empire’s Prime Number Checker, you will see that it is prime.
Thus 811 is the smaller of 24!+1’s two prime factors!
Print the puzzles or type the solution on this excel file: 10-factors 807-814
One of my education professors taught that you can teach any concept with a picture book.
I recently read the book, Stick and Stone, to a class of 6th graders. Yes, 6th graders. You can get away with reading something way below grade level if you tell them before you start reading that you will use the book to introduce them to something that is definitely NOT below grade level. The first few pages of the book are shared by its publisher here:
As you can see, those first few pages equate stone as a zero and stick as a lonely number one.
The middle part of the book teaches about synergizing, working together to make life good and helping each other through tough times.
By the end of the book Stick and Stone know how to work very well together, “Stick, Stone. Together again. Stick, Stone. A perfect ten.”
The book pretty much ends there, but making a perfect ten is only the beginning of what these two characters can do together. I used this book to teach the class not only about getting along and working together, but also about base 2, or binary as it is also called. Every counting number we know can be represented by using just 1’s and 0’s. I wrote on the board the numbers from 1 to 16 and represented the first few of those numbers in base 2. Then I invited class members to come up with how to write the rest of the numbers in base 2. Some students caught on immediately while the others were able to learn how to do it by watching their classmates and listening to them. Eventually with at least 12 different student’s inputs, we came up with a chart that looked something like this:
Notice that the numbers from 9 to 15 are just 1000 plus the numbers directly across from them in the first column.
Some of the sixth grade students had already heard of binary, so I showed them a little more about base 2: I wrote a bunch of 1’s and 0’s “off the top of my head” onto the board and added the headings to show place values: 1’s place, 2’s place, etc.
Then I told them to sum up the place values that contained a one:
The sixth graders were delighted with the answer.
Stick and Stone are the main two characters, but the book has one other character, Pinecone. At first Pinecone bullied Stone, but after Stick stood up to him, the three of them were eventually able to become friends. You might enjoy finding out more about Pinecone by listening to Sean Anderson read the entire book to his children, one of which seems to really enjoy numbers.
Children might also enjoy representing all the numbers from 1 to 31 with one hand:
If you used a unique symbol to represent Pinecone, it could look like a 2. Then you also could use the symbols 0, 1, and 2 to represent every counting number in base 3. That’s another concept the picture book Stick and Stone could be used to introduce!
To make a chart for base 3, start with these 3 columns of numbers with 3 numbers in each:
Since this is base 3, where should 10 and 100 go? The bottom of the first column and the bottom of the third column both MUST look like a power of 10. The rest of the chart is easy to fill out. Notice the 1 and 2 look exactly the same in base 10 and base 3. Also since 4 = 3 + 1, 5 = 3 + 2, and 6 = 3 + 3, we can easily fill in the 2nd column. Two more addition facts will finish the third column: 7 = 6 + 1, and 8 = 6 + 2.
Now add what you learned about 4, 5, 6, 7, 8, and 9 to column 1 and put the numbers 10 – 18 in the base 10 second column and numbers 19 – 27 in the base 10 third column. Again the bottom of the first column and the bottom of the third column both MUST look like a power of 10, so we now know where to put 1000.
To fill in the rest of the chart simply add 100 to the base 3 numbers in column 1 to get the the base 3 numbers in column 2. Then add 200 to the base 3 numbers in column 1 to get the remaining base 3 numbers in column 3.
You could do this process again to determine the first 81 counting numbers in base 3 with 81 being represented by 10000.
For base 4, you could do something similar with 4 columns. However, for counting in bases 4, 5, 6, 7, 8, and 9 I would suggest using the very versatile hundred chart. You can give instructions without even mentioning the concept of differing bases. For example, cross out every number on the hundred chart that has 7, 8, or 9 as one or more of its digits. Can you tell even before you get started how many numbers will get crossed out? (100 – 7²) What pattern do the cross-outs make? If you arrange the remaining numbers in order from smallest to largest, then you will have the first 49 numbers represented in base 7. With a minimal amount of cutting and taping you could have a “hundred” chart in base 7. Easy peasy.
This excel file not only has several puzzles, including today’s, but also a hundred chart and even a thousand chart because I know some of you might want to play with 3-digit numbers, too.
Now let me tell you a little bit about the number 810:
810 is a composite number.
Prime factorization: 810 = 2 x 3 x 3 x 3 x 3 x 5, which can be written 810 = 2 × 3⁴ × 5
The exponents in the prime factorization are 1, 4 and 1. Adding one to each and multiplying we get (1 + 1)(4 + 1)(1 + 1) = 2 x 5 x 2 = 20. Therefore 810 has exactly 20 factors.
Factor pairs: 810 = 1 x 810, 2 x 405, 3 x 270, 5 x 162, 6 x 135, 9 x 90, 10 x 81, 15 x 54, 18 x 45 or 27 x 30
Taking the factor pair with the largest square number factor, we get √810 = (√81)(√10) = 9√10 ≈ 28.4604989.
Since 810 has so many factors, it has MANY possible factor trees. If most people made a factor tree for 810, they would probably start with 81 × 10 or 9 x 90. NOT ME! Here are two less-often-used factor trees for 810:
Finally, here is an easy way to express 810 is in a different base:
Make a cake in which you divide 810 by the base number repeatedly, keeping track of the remainders, including zero, as you go.
Keep dividing until the number at the top of the cake is 0.
List the remainders in order from top to bottom and indicate the base you used to do the division.
This method is illustrated for BASE 2 and BASE 3 below:
That’s all pretty good work for a stone, a stick, and a pine cone!
By the way, using that method will also produce the following results:
810 is 30222 BASE 4
810 is 11220 BASE 5
810 is 3430 BASE 6 and so forth.
And just so you’ll know, 810 is the sum of consecutive primes 401 and 409.
