Eight is half of sixteen, so 816 is divisible by 6. You probably weren’t expecting that divisibility rule, but it’s true.

816 can also be easily divided by 2, 4, and 8. How many factors does 816 have in all? Plenty! Scroll down past the puzzle and see!

816 is a composite number.

Prime factorization: 816 = 2 x 2 x 2 x 2 x 3 x 17, which can be written 816 = (2^4) x 3 x 17

The exponents in the prime factorization are 4, 1 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 x 2 x 2 = 20. Therefore 816 has exactly 20 factors.

Since multi-digit 815 ends with 5, it is a composite number, and it is also the hypotenuse of a Pythagorean triple:

489-652-815 which is 163 times 3-4-5.

Can you write the numbers 1 – 12 in both the first column and the top row so that this puzzle functions like a multiplication table?

815 is a composite number.

Prime factorization: 815 = 5 x 163

The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 815 has exactly 4 factors.

Factors of 815: 1, 5, 163, 815

Factor pairs: 815 = 1 x 815 or 5 x 163

815 has no square factors that allow its square root to be simplified. √815 ≈ 5482048.

Here she is sitting with my husband, me, and her big sister. My husband and I are pretty new to her so she’s probably thinking in Mandarin, “Who are these people?” On the other hand, we are delighted to be a part of her life now.

Here is today’s puzzle:

Print the puzzles or type the solution on this excel file: 10-factors 807-814

Here’s a little about the number 813:

813 is a composite number.

Prime factorization: 813 = 3 x 271

The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 813 has exactly 4 factors.

Factors of 813: 1, 3, 271, 813

Factor pairs: 813 = 1 x 813 or 3 x 271

813 has no square factors that allow its square root to be simplified. √813 ≈ 51315.

813 is repdigit 111 in BASE 28 because 28² + 28¹ + 28º = 813

Stetson.edu and my calculator informed me that 813^e = 81,366,615.06223032 . . .

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 to be 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 = (2^2) 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.

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.

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 x x 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:

Print the puzzles or type the solution on this excel file: 10-factors 807-814

Normally I would tell you that 809 is a palindrome in two different bases:

676 in BASE 11 because 6(121) + 7(11) + 6(1) = 809

575 in BASE 12 because 5(144) + 7(12) +5(1) = 809

But whoop-de-doo, all that really means is that (x – 11) is a factor of 6x² + 7x – 803, and (x – 12) is a factor of 5x² + 7x – 804.

Isn’t it just as exciting that

(x – 2) is a factor of x^9 + x^8 + x^5 + x³ – 808 because 809 is 1100101001 in BASE 2?

(x – 3) is a factor of x^6 + 2x³ + 2x² +2x – 807 because 809 is 1002222 in BASE 3?

(x – 4) is a factor of 3x^4 + 2x² + 2x – 808 because 809 is 30221 in BASE 4?

(x – 5) is a factor of x^4 + x³ + 2x² + x – 805 because 809 is 11214 in BASE 5?

Notice that the last number in each of those polynomials is divisible by the BASE number.

Palindromes NEVER end in zero so the polynomials they produce will NEVER end in the original base 10 number.

So are palindromes really so special? Today I am much more excited that figuring out what a number is in another base can give us a factor of a corresponding polynomial!

How do I know what those polynomials are? Let me use 809 in BASE 6 as an example:

Since 809 is 3425 in BASE 6, I know that

3(6³) + 4(6²) + 2(6¹) + 5(6º) = 809

3(216) + 4(36) + 2(6) + 5(1) – 809 = 0

so 3(216) + 4(36) + 2(6) – 804 = 0

thus (x – 6) is a factor of 3x³ + 4x² + 2x – 804 because of the factor theorem.

If I told you what 809 is in Bases 7, 8, 9, and 10 would you be able to write the corresponding polynomials that are divisible by (x – 7), (x – 8), (x – 9), and (x – 10) respectively?

2234 in BASE 7

1451 in BASE 8

1088 in BASE 9

809 in BASE 10

Scroll down past 809’s factoring information to see if you found the correct polynomials.

—————–

809 is a prime number.

Prime factorization: 809 is prime and cannot be factored.

The exponent of prime number 809 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 809 has exactly 2 factors.

Factors of 809: 1, 809

Factor pairs: 809 = 1 x 809

809 has no square factors that allow its square root to be simplified. √809 ≈ 28.4429253066558.

How do we know that 809 is a prime number? If 809 were not a prime number, then it would be divisible by at least one prime number less than or equal to √809 ≈ 28.4. Since 809 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 809 is a prime number.

Here’s another way we know that 809 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 28² + 5² = 809 with 28 and 5 having no common prime factors, 809 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √809 ≈ 28.4. Since 809 is not divisible by 5, 13, or 17, we know that 809 is a prime number.

——————–

Were you able to find those polynomials from knowing what 809 is in other bases? Check your work with the answers below:

2234 Base 7 tells us (x – 7) is a factor of 2x³ + 2x² + 3x – 805

1451 Base 8 tells us (x – 8) is a factor of x³ + 4x² + 5x – 808

1088 Base 9 tells us (x – 9) is a factor of x³ + 8x – 801

809 Base 10 tells us (x – 10) is a factor of 8x² – 800

If you’ve made it this far, even if I’ve made you feel a little dizzy, you’ve done GREAT! Keep up the good work!

Happy birthday to my good friend, Justin! He seems to always remember the birthdays of everyone he knows, so this is how I am remembering his special day today. Justin is highly intelligent, thoughtful, and very friendly. I am confident he can solve this Level 6 puzzle that looks a little like a birthday cake.

Print the puzzles or type the solution on this excel file: 10-factors 807-814

This is my 808th post so I thought I would also make a factor cake for the number 808. It’s prime factor, 101, is at the top of the cake. Justin, I hope you live to be 101!

808 is a palindrome. That means it looks the same forwards and backwards. It is also a strobogrammatic number. That means it looks the same right side up or upside down.

ALL of the factors of 808 are also palindromes, and four of them are strobogrammatic numbers, too. Can you figure out which ones are both?

808 is a composite number.

Prime factorization: 808 = 2 x 2 x 2 x 101, which can be written 808 = (2^3) x 101

The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 x 2 = 8. Therefore 808 has exactly 8 factors.

Factors of 808: 1, 2, 4, 8, 101, 202, 404, 808

Factor pairs: 808 = 1 x 808, 2 x 404, 4 x 202, or 8 x 101

Taking the factor pair with the largest square number factor, we get √808 = (√4)(√202) = 2√202 ≈ 428.425340807

Here are the factors that make puzzle #808 act like a multiplication table. It is followed by a table of logical steps to arrive at that solution.

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