Level 6 Puzzle

834 and Level 6

/ ivasallay / Leave a comment

834 is the sum of consecutive prime numbers two different ways:

  • 127 + 131 + 137 + 139 + 149 + 151 = 834; that’s six consecutive primes
  • 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 = 834; that’s fourteen consecutive primes

The ONLY Pythagorean triple that contains the number 834 is 834 – 173888 – 173890.

  • 834 is a composite number.
  • Prime factorization: 834 = 2 × 3 × 139
  • 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 × 2 × 2 = 8. Therefore 834 has exactly 8 factors.
  • Factors of 834: 1, 2, 3, 6, 139, 278, 417, 834
  • Factor pairs: 834 = 1 × 834, 2 × 417, 3 × 278, or 6 × 139
  • 834 has no square factors that allow its square root to be simplified. √834 ≈ 28.879058

There was a solar eclipse in the United States today. People where I lived were able to experience 91.32% obstruction of the sun. I love this interactive map of today’s eclipse and past and future ones as well.

Several people have taken and shared marvelous pictures of the eclipse.

Here are a few tweets I saw about eclipses on twitter:

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828 Try Synthetic Division on These Special Polynomials

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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.
  • Factors of 828: 1, 2, 3, 4, 6, 9, 12, 18, 23, 36, 46, 69, 92, 138, 207, 276, 414, 828
  • Factor pairs: 828 = 1 × 828, 2 × 414, 3 × 276, 4 × 207, 6 × 138, 9 × 92, 12 × 69, 18 × 46 or 23 × 36
  • 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.

 

828 Synthetic Division

make science GIFs like this at MakeaGif

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:

  1. 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.
  2. 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.
  3. 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.
  4. 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 up to 820 with Exactly 12 Factors

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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.
  • Factors of 820: 1, 2, 4, 5, 10, 20, 41, 82, 164, 205, 410, 820
  • Factor pairs: 820 = 1 × 820, 2 × 410, 4 × 205, 5 × 164, 10 × 82, or 20 × 41
  • 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.

808 Happy Birthday, Justin!

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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 ≈ 28.425340807

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.

806 and a Level 6 Dunce Cap?

/ ivasallay / 2 Comments

When I put this post together I took a second look at today’s puzzle and thought, “That looks a little like a dunce cap.” That thought led me to two very interesting articles whose information surprised me greatly.

The first one titled “The Dunce Cap Wasn’t Always so Stupid” explains that long ago when the dunce cap was first introduced by the brilliant Scotsman John Duns Scotus, it became a symbol of exceptional intellect. In fact wizard hats were most likely modeled after them. Unfortunately, this positive perception of the caps remained for only a couple of centuries.

The second article is short but helped me visualize Topology’s Dunce Hat. I enjoyed watching the animation of this mathematical concept.

I hope you will enjoy trying to solve the puzzle. It is a Level 6, so it won’t be easy. If you succeed, you’ll deserve to feel that you have exceptional intellect.

Print the puzzles or type the solution on this excel file: 10-factors 801-806

Now here is some information about the number 806:

  • 806 is a composite number.
  • Prime factorization: 806 = 2 x 13 x 31
  • 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 806 has exactly 8 factors.
  • Factors of 806: 1, 2, 13, 26, 31, 62, 403, 806
  • Factor pairs: 806 = 1 x 806, 2 x 403, 13 x 62, or 26 x 31
  • 806 has no square factors that allow its square root to be simplified. √806 ≈ 28.390139.

806 is a palindrome in three different bases:

  • 11211 BASE 5 because 1(625) + 1(125) + 2(25) + 1(5) + 1(1) = 806
  • 1C1 BASE 23 (C is 12 base 10) because 1(23²) + 12(23) + 1(1) = 806
  • QQ BASE 30 (Q is 26 base 10) because 26(30) + 26(1) = 806, which follows naturally from the fact that 26 × 31 = 806

806 is the hypotenuse of Pythagorean triple 310-744-806 which is 5-12-13 times 62.

And 806 can be written as the sum of three squares seven different ways:

  • 26² + 11² + 3² = 806
  • 26² + 9² + 7² = 806
  • 25² + 10² + 9² = 806
  • 23² + 14² + 9² = 806
  • 21² + 19² + 2² = 806
  • 21² + 14² + 13² = 806
  • 19² + 18² + 11² = 806

 

799 A Rose for Your Valentine

/ ivasallay / 1 Comment

Roses are beautiful and make lovely gifts for Valentines or any other occasion. A Native American legend explains why roses have thorns.

The rose on today’s puzzle has thorns because without thorny clue 60 the puzzle would not have a unique solution. You can be sure that 60 will play an important part in using logic to find the solution to this puzzle.

With or without a valentine, love your brain and give the puzzle a try. It won’t be easy, but you should eventually be able to figure it out. My blogging friend, justkinga, has some other suggestions to show YOURSELF some love on Valentine’s Day.

799-puzzle

Print the puzzles or type the solution on this excel file: 12-factors-795-799

7 + 9 + 9 = 25, a composite number.

7^3 + 9^3 + 9^3 = 1801, a prime number.

OEIS.org states that 799 is the smallest number whose digits add up to a composite number AND whose digits cubed add up to a prime number.

It may seem like an improbable number fact, but it wasn’t too difficult to verify, and it really is true!

799 is also the smallest number whose digits add up to 25. (The digits of 889 also add up to 25, and its digits cubed also add up to a prime number. Could this be more than a coincidence?)

Here’s more about the number 799:

799 is palindrome 1H1 in BASE 21 (H is 17 base 10). Note that 1(441) + 17(21) + 1(1) = 799.

799 is the hypotenuse of Pythagorean triple 376-705-799 which is 47 times 8-15-17.

  • 799 is a composite number.
  • Prime factorization: 799 = 17 x 47
  • 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 799 has exactly 4 factors.
  • Factors of 799: 1, 17, 47, 799
  • Factor pairs: 799 = 1 x 799 or 17 x 47
  • 799 has no square factors that allow its square root to be simplified. √799 ≈ 266588.

799-factor-pairs

 

 

 

794 and Level 6

/ ivasallay / Leave a comment

794 is the hypotenuse of a Pythagorean triple, 456-650-794, so 456² + 650² = 794².

794 is also palindrome 282 in BASE 18. Note that 2(18²) + 8(18) + 2(1) = 794.

OEIS.org informs us that 1⁶ + 2⁶ + 3⁶ = 1 + 64 + 729 = 794.

794-puzzle

Print the puzzles or type the solution on this excel file: 10-factors-788-794

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  • 794 is a composite number.
  • Prime factorization: 794 = 2 x 397
  • 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 794 has exactly 4 factors.
  • Factors of 794: 1, 2, 397, 794
  • Factor pairs: 794 = 1 x 794 or 2 x 397
  • 794 has no square factors that allow its square root to be simplified. √794 ≈ 1780056.

794-factor-pairs

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787 Always a Unique Solution

/ ivasallay / Leave a comment
  • 787 is a prime number.
  • Prime factorization: 787 is prime.
  • The exponent of prime number 787 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 787 has exactly 2 factors.
  • Factors of 787: 1, 787
  • Factor pairs: 787 = 1 x 787
  • 787 has no square factors that allow its square root to be simplified. √787 ≈ 28.05352.

787-factor-pairs

Now for today’s puzzle….The fact that these Find the Factor puzzles always have a unique solution is an important clue in solving this rather difficult puzzle. Good luck!

787-puzzle

Print the puzzles or type the solution on this excel file: 12-factors-782-787

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Here’s more about the number 787:

787 is a palindrome in bases 4, 10, 11 and 16:

  • 30103 BASE 4; note that 3(256) + 0(64) + 1(16) + 0(4) + 3(1) = 787
  • 787 BASE 10; note that 7(100) + 8(10) + 7(1) = 787
  • 656 BASE 11; note that 6(121) + 5(11) + 6(1) = 787
  • 313 BASE 16; note that 3(256) + 1(16) + 3(1) = 787

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What did I mean when I wrote that the puzzles always having a unique solution is an important clue? There is only one clue in the puzzle that is divisible by 11. One of the rows and one of the columns do not have a clue, so the other 11 will go with one of them. The cell where the empty row and empty column intersect cannot be 132 because if that worked, it would produce two possible solutions to the puzzle. This table explains a logical order to find the solution.

787-logic

781 and Level 6

/ ivasallay / Leave a comment
  • 781 is a composite number.
  • Prime factorization: 781 = 11 x 71
  • 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 781 has exactly 4 factors.
  • Factors of 781: 1, 11, 71, 781
  • Factor pairs: 781 = 1 x 781 or 11 x 71
  • 781 has no square factors that allow its square root to be simplified. √781 ≈ 27.94637722.

781-factor-pairs

Can you solve today’s puzzle?

781-puzzle

Print the puzzles or type the solution on this excel file: 10-factors-2016

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Here’s more about the number 781:

781 is the sum of the 19 prime numbers from 7 to 79.

Thus 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 = 781

781 is a repdigit in base 5 and a palindrome in three other bases:

  • 11111 BASE 5. Note that 625 + 125 + 25 + 5 + 1 = 781.
  • 232 BASE 19. Note that 2(19²) + 3(19) + 2(1) = 781
  • 1J1 BASE 20, J = 19 base 10. Note that 1(20²) + 19(20) + 1(1) = 781
  • 141 BASE 26. Note that 1(26²) + 4(26) + 1(1) = 781

781 is also the sum of three squares five ways

  • 27² + 6² + 4² = 781
  • 24² + 14² + 3² = 781
  • 24² + 13² + 6² = 781
  • 21² + 18² + 4² = 781
  • 21² + 14² + 12² = 781

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781-logic

773 and Level 6

/ ivasallay / 2 Comments
  • 773 is a prime number.
  • Prime factorization: 773 is prime.
  • The exponent of prime number 773 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 773 has exactly 2 factors.
  • Factors of 773: 1, 773
  • Factor pairs: 773 = 1 x 773
  • 773 has no square factors that allow its square root to be simplified. √773 ≈ 27.8028775.

773-factor-pairs

How do we know that 773 is a prime number? If 773 were not a prime number, then it would be divisible by at least one prime number less than or equal to √773 ≈ 27.8. Since 773 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 773 is a prime number.

Here is today’s puzzle for you to try to solve:

773 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2016-02-25

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What else is special about the number 773?

22² + 17² = 773 so 773 is the hypotenuse of the primitive Pythagorean triple 195-748-773 which was calculated using 22² – 17², 2(17)(22), 22² + 17².

Thus 195² + 748² + 773².

773 is also the sum of three squares six different ways:

  • 26² + 9² + 4² = 773
  • 25² + 12² + 2² = 773
  • 24² + 14² + 1² = 773
  • 23² + 12² + 10² = 773
  • 22² + 15² + 8² = 773
  • 20² + 18² + 7² = 773

773 is a palindrome in two other bases:

  • 545 BASE 12, note that 5(144) + 4(12) + 5(1) = 773
  • 3D3 BASE 14 (D = 13 base 10); note that 3(196) + 13(14) + 3(1) = 773

Here’s another way we know that 773 is a prime number: Since  its last two digits divided by 4 leave a remainder of 1, and 22² + 17² = 773 with 22 and 17 having no common prime factors, 773 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √773 ≈ 27.8. Since 773 is not divisible by 5, 13, or 17, we know that 773 is a prime number.

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773 Logic