## A Multiplication Based Logic Puzzle

### 797 and Mathematical Ways to Love

Mathwithbaddrawings.com has some thoughtful and entertaining Ways to Tell a Mathematician that you love them.

Artful Maths wrote a post that includes beautiful mathematical origami valentines and a “string art” cardioid that is made with a pencil instead of string.

This puzzle could be another mathematical way to show some love:

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

Here are a few facts about the number 797:

797 is a palindrome in three bases:

• 797 BASE 10 because 7(100) + 9(10) + 7(1) = 797
• 565 BASE 12 because 5(144) + 6(12) + 5(1) = 797
• 494 BASE 13 because 4(169) + 9(13) + 4(1) = 797

But there’s one more palindromic fact about the number 797: It is the sum of two square numbers that are also palindromes!

• 797 = 121 + 676. Note that 11² = 121 and 26² = 676.

Since it is the sum of two squares, 797 will also be the hypotenuse a Pythagorean triple:

• 555 – 572 – 797 calculated from 26² – 11², 2(26)(11), 26² + 11².

797 is the sum of three squares seven different ways:

• 28² + 3² + 2² = 797
• 27² + 8² + 2² = 797
• 24² + 14² + 5² = 797
• 24² + 11² + 10² = 797
• 22² + 13² + 12² = 797
• 21² + 16² + 10² = 797
• 20² + 19² + 6² = 797

797 is also the sum of the 15 prime numbers from 23 to 83:

• 23 + 29 + 31 + 37+ 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 = 797

Here is the most basic information about the number 797:

• 797 is a prime number.
• Prime factorization: 797 is prime and cannot be factored.
• The exponent of prime number 797 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 797 has exactly 2 factors.
• Factors of 797: 1, 797
• Factor pairs: 797 = 1 x 797
• 797 has no square factors that allow its square root to be simplified. √797 ≈ 28.231188.

How do we know that 797 is a prime number? If 797 were not a prime number, then it would be divisible by at least one prime number less than or equal to √797 ≈ 28.2. Since 797 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 797 is a prime number.

Here’s another way we know that 797 is a prime number: Since  its last two digits divided by 4 leave a remainder of 1, and 26² + 11² = 797 with 26 and 11 having no common prime factors, 797 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √797 ≈ 28.2. Since 797 is not divisible by 5, 13, or 17, we know that 797 is a prime number.

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### 727 Enjoy Some Sparkling Cider!

Have a happy New Year’s Eve! My husband and I will be enjoying some Sparkling Apple-Pear. I invite you to have some, too.

Perhaps the goblet in the puzzle can start off your festivities.

Print the puzzles or type the solution on this excel file: 12 Factors 2015-12-28

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• 727 is a prime number.
• Prime factorization: 727 is prime and cannot be factored.
• The exponent of prime number 727 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 727 has exactly 2 factors.
• Factors of 727: 1, 727
• Factor pairs: 727 = 1 x 727
• 727 has no square factors that allow its square root to be simplified. √727 ≈ 26.9629375.

727 is a palindrome in base 10 and one other base:

• 727 BASE 10; note that 7(100) + 2(10) + 7(1) = 727.
• 1B1 BASE 22 (B = 11 base 10); note that 1(22²) + 11(22) + 1(1) = 727.

Stetson.edu informs us that 727² = 528529, a number whose digits can be split in half to make two consecutive numbers.

Since 727 is a prime number, there is only one way it can be expressed as the sum of consecutive numbers: 363 + 364 = 727.

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### When is 690 a Palindrome? In Base 16 and Base 29.

• 690 is a composite number.
• Prime factorization: 690 = 2 x 3 x 5 x 23
• The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 x 2 = 16. Therefore 690 has exactly 16 factors.
• Factors of 690: 1, 2, 3, 5, 6, 10, 15, 23, 30, 46, 69, 115, 138, 230, 345, 690
• Factor pairs: 690 = 1 x 690, 2 x 345, 3 x 230, 5 x 138, 6 x 115, 10 x 69, 15 x 46, or 23 x 30
• 690 has no square factors that allow its square root to be simplified. √690 ≈ 26.267851

Here is today’s factoring puzzle:

Print the puzzles or type the solution on this excel file: 10 Factors 2015-11-23

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Here is a little more about the number 690:

690 is the sum of the six prime numbers from 103 to 131. Do you know what all of those prime numbers are?

690 is also the hypotenuse of Pythagorean triple 414-552-690. What is the greatest common factor of those three numbers?

In BASE 10 we use the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. For bases smaller than 10 we use only the digits less than that base. For bases greater than 10, we might use A to represent 10, B to represent 11, and so forth all the way to Z representing 35 in BASE 36.

690 is a palindrome in two bases that require us to use letters of the alphabet to represent it:

• 2B2 in BASE 16; note that 2(256) + 11(16) + 2(1) = 690
• NN in BASE 29; note that 23(29) + 23(1) = 690. (N is the 14th letter of the alphabet and 14 + 9 = 23)

NN looks like it is divisible by 11, but remember that 11 base 29 is the same as 30 in base 10.

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### 686 and Level 5

686 is divisible by 2, not by 3, and not by 5. Is it divisible by 7? Let’s try a divisibility rule for 7’s: 68 – 2(6) = 56, a multiple of 7. Yes, 686 is divisible by 7.

686 is a palindrome in two bases:

• 686 in base 10; note that 6(100) + 8(10) + 6(1) = 681
• 222 in base 18; note that 2(324) + 2(18) + 2(1) = 681

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-16

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• 686 is a composite number.
• Prime factorization: 686 = 2 x 7 x 7 x 7, which can be written 686 = 2 x (7^3)
• The exponents in the prime factorization are 1 and 3. Adding one to each and multiplying we get (1 + 1)(3 + 1) = 2 x 4 = 8. Therefore 686 has exactly 8 factors.
• Factors of 686: 1, 2, 7, 14, 49, 98, 343, 686
• Factor pairs: 686 = 1 x 686, 2 x 343, 7 x 98, or 14 x 49
• Taking the factor pair with the largest square number factor, we get √686 = (√49)(√14) = 7√14 ≈ 26.1916017.

Here is a little cake to simplify √686:

Just take the square root of everything on the outside of the cake and multiply them together to get √686 = (√2)(√7)(√49) = 7√14

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### 626 Semordnilaps, Palindromes and Level 4

626 is the hypotenuse of the Pythagorean triple 50-624-626. What is the greatest common factor of those three numbers?

Today my son posted on facebook, “A word that when spelled backwards spells a different word is called a Semordnilap.” That’s a word I hadn’t heard before, and it also applies to phrases and sentences that form different phrases or sentences when read backwards.

“Semordnilap” is a semordnilap for the word “palindromes” which are words, phrases, sentences, and numbers that read the same forward and backward.

Palindrome sentences are sometimes made with words that are semordnilaps: was, saw, live, evil, on, no, desserts, stressed, stop, pots, tops, spot, diaper, repaid.

626 is a number that is a palindrome in several different bases:

• 10001 in base 5; note that (5^4) + 1 = 626.
• 626 in base 10
• 272 in base 16; note that 2(16^2) + 7(16) + 2 = 626
• 1DI in base 19, if “1” and “I” look the same, and too often they do. Note that 1(19^2) + 13(19) + 18 = 626
• 101 in base 25; note that (25^2) + 1 = 626
• 11 in base 625; note that 626 + 1 = 626

I guess we could say that in all other bases 626 is a semordnilap.

All of 626’s factors are palindromes, too.

Print the puzzles or type the solution on this excel file: 12 Factors 2015-09-21

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• 626 is a composite number.
• Prime factorization: 626 = 2 x 313
• 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 626 has exactly 4 factors.
• Factors of 626: 1, 2, 313, 626
• Factor pairs: 626 = 1 x 626 or 2 x 313
• 626 has no square factors that allow its square root to be simplified. √626 ≈ 25.019992.

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### What can you see in the number 585?

This week I watched an excellent video titled 5 x 9 is more than 45. Indeed 45 is so much more than simply 5 x 9. Every multiplication fact is much more than that mere fact, but Steve Wyborney used 5 x 9 = 45 in his video… Guess what! 585 is a multiple of 45.

As I thought about the number 585, I marveled at some of the hidden mysteries this number holds.

Since 585 is divisible by two different centered square numbers, 5 and 13, I saw that 585 could be represented by this lovely array that has 45 larger squares made up of 13 smaller colorful squares. When you look at the array, do you just see 585 squares or can you see even more multiplication and division facts? If you rotate the array 90 degrees, do the facts change?

What do you see in this array of 117 medium sized squares made up of 5 smaller squares:

Or this more simple looking array of sixty-five 3 x 3 squares.

All of these arrays are in just two dimensions. A 5 x 9 x 13 rectangular prism is ONE way to represent 585 in three dimensions.

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Since factors 5, 13, and 65 are hypotenuses of primitive Pythagorean triples, 585 is the hypotenuse of four Pythagorean triples. Each triple has a different greatest common factor. Can you figure out what each one is?

• 144-567-585
• 225-540-585
• 297-504-585
• 351-468-585

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And if all of that wasn’t enough, Stetson.edu informs us that 585 is a palindrome in 3 different bases!

• 585 = 1001001001 in base 2
• 585 = 1111 in base 8
• 585 = 585 in base 10

Here are two different methods of determining 585 in base 8. In both methods the base 8 representation is in blue.

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• 585 is a composite number.
• Prime factorization: 585 = 3 x 3 x 5 x 13, which can be written 585 = (3^2) x 5 x 13
• 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 585 has exactly 12 factors.
• Factors of 585: 1, 3, 5, 9, 13, 15, 39, 45, 65, 117, 195, 585
• Factor pairs: 585 = 1 x 585, 3 x 195, 5 x 117, 9 x 65, 13 x 45, or 15 x 39
• Taking the factor pair with the largest square number factor, we get √585 = (√9)(√65) = 3√65 ≈ 24.18677