Here are two puzzles that go together and yet look out of sync. Sometimes cupid’s arrow reaches its target, and sometimes it doesn’t.

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

Here’s a little about the number 798:

798 is made from three consecutive numbers (7, 8, and 9), so it is divisible by three. The middle number, 8, is not divisible by three, so 798 is NOT divisible by nine.

798 is a palindrome in two bases:

• 666 BASE 11 because 6(121) + 6(11) + 6(1) = 798
• 383 BASE 15 because 3(225) + 8(15) + 3(1) = 798

798 is also the sum of two consecutive prime numbers: 397 + 401 = 798.

798 can be written as the sum of three squares four different ways:

• 26² + 11² + 1² = 798
• 25² + 13² + 2² =798
• 23² + 13² + 10² = 798
• 22² + 17² + 5² = 798

Here is 798’s factoring information:

• 798 is a composite number.
• Prime factorization: 798 = 2 x 3 x 7 x 19
• 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 798 has exactly 16 factors.
• Factors of 798: 1, 2, 3, 6, 7, 14, 19, 21, 38, 42, 57, 114, 133, 266, 399, 798
• Factor pairs: 798 = 1 x 798, 2 x 399, 3 x 266, 6 x 133, 7 x 114, 14 x 57, 19 x 42, or 21 x 38
• 798 has no square factors that allow its square root to be simplified. √798 ≈ 28.24889.

I learned yesterday that pushing ALT on the computer key pad and 3 on the number pad at the same time produces a ♥ when the keys are released. I added a few ♥‘s to today’s puzzle because Valentine’s day is almost here:

• 795 is a composite number.
• Prime factorization: 795 = 3 x 5 x 53
• 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 795 has exactly 8 factors.
• Factors of 795: 1, 3, 5, 15, 53, 159, 265, 795
• Factor pairs: 795 = 1 x 795, 3 x 265, 5 x 159, or 15 x 53
• 795 has no square factors that allow its square root to be simplified. √795 ≈ 28.195744.

OEIS.org informs us that if we add up all the factors of 795 we get a number that is a perfect 4th power:

• 1 + 3 + 5 + 15 + 53 + 159 + 265 + 795 = 1296, which happens to be the 4th power of 6. (Of course 1296 also equals 36 squared.)

795 is the hypotenuse of four Pythagorean triples and thus:

• 69 – 792 – 795 which is 3 times (23 – 264 – 265)
• 288 – 741 – 795 which is 3 times (96 – 247 – 265)
• 420 – 675 – 795 which is 15 times (28 – 45 – 53)
• 477 – 636 – 795 which is 159 times (3 – 4 – 5)

795 is also the sum of three squares two different ways:

• 25² + 13² + 1² = 795
• 25² + 11² + 7² = 795

To see if 791 is divisible by 7, you could try either one of these divisibility tricks:

• 791 is divisible by 7 because 79 – 2(1) = 77 which obviously is divisible by 7.
• 791 is divisible by 7 because 79 + 5(1) = 84 which most people know is 12 × 7.

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

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• 791 is a composite number.
• Prime factorization: 791 = 7 x 113
• 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 791 has exactly 4 factors.
• Factors of 791: 1, 7, 113, 791
• Factor pairs: 791 = 1 x 791 or 7 x 113
• 791 has no square factors that allow its square root to be simplified. √791 ≈ 28.12472222.

791 is the hypotenuse of Pythagorean triple 105-784-791 which is 15-112-113 times 7.

791 is also the sum of seven consecutive prime numbers:

• 101 + 103 + 107 + 109 + 113 + 127 + 131 = 791

• 790 is a composite number.
• Prime factorization: 790 = 2 x 5 x 79
• 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 790 has exactly 8 factors.
• Factors of 790: 1, 2, 5, 10, 79, 158, 395, 790
• Factor pairs: 790 = 1 x 790, 2 x 395, 5 x 158, or 10 x 79
• 790 has no square factors that allow its square root to be simplified. √790 ≈ 28.106939.

Here is today’s puzzle:

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

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Here’s a little more about the number 790:

There are four ways to make 790 using three squares:

• 21² + 18² + 5² = 790
• 27² + 6² + 5² = 790
• 23² + 15² + 6² = 790
• 22² + 15² + 9² = 790

790 is the hypotenuse of Pythagorean triple 474-632-790 which is 3-4-5 times 158.

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789 consists of exactly three consecutive numbers so it is divisible by 3.

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

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• 789 is a composite number.
• Prime factorization: 789 = 3 x 263
• 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 789 has exactly 4 factors.
• Factors of 789: 1, 3, 263, 789
• Factor pairs: 789 = 1 x 789 or 3 x 263
• 789 has no square factors that allow its square root to be simplified. √789 ≈ 28.08914.

789 is the sum of consecutive prime numbers 2 different ways:

• 257 + 263 + 269 = 789 (that’s 3 consecutive primes.)
• 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 = 789 (that’s 21 consecutive primes!)

789 is the sum of three squares eight different ways:

• 28² + 2² + 1² = 789
• 26² + 8² + 7² = 789
• 25² + 10² + 8² = 789
• 23² + 16² + 2² = 789
• 23² + 14² + 8² = 789
• 22² + 17² + 4² = 789
• 22² + 16² + 7² = 789
• 20² + 17² + 10² = 789

786 is even so it is divisible by 2. Also since 786 is made from 3 consecutive numbers, we can tell automatically that it is divisible by 3. Those two facts together mean 786 is also divisible by 6.

• 786 is a composite number.
• Prime factorization: 786 = 2 x 3 x 131
• 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 786 has exactly 8 factors.
• Factors of 786: 1, 2, 3, 6, 131, 262, 393, 786
• Factor pairs: 786 = 1 x 786, 2 x 393, 3 x 262, or 6 x 131
• 786 has no square factors that allow its square root to be simplified. √786 ≈ 28.03569.

Today’s Find the Factors puzzle:

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

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Here’s a little more about the number 786:

786 is 123 in BASE 27 because 1(27²) + 2(27) + 3(1) = 786.

786 is the sum of two consecutive primes: 389 + 397 = 786

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If you know the multiplication facts up to 12 x 12, then it is obvious that 784 is divisible by 7. However, 784 is divisible by a whole lot more numbers than that. 784 can actually be evenly divided by 15 different numbers. Hmm, 15, that’s an odd number. A number’s factors always come in pairs. One of its factor pairs must contain the same factor twice, meaning 784 is a perfect square!

• 784 is a composite number.
• Prime factorization: 784 = 2 x 2 x 2 x 2 x 7 x 7, which can be written 784 = (2^4) x (7^2)
• The exponents in the prime factorization are 4 and 2. Adding one to each and multiplying we get (4 + 1)(2 + 1) = 5 x 3 = 15. Therefore 784 has exactly 15 factors.
• Factors of 784: 1, 2, 4, 7, 8, 14, 16, 28, 49, 56, 98, 112, 196, 392, 784
• Factor pairs: 784 = 1 x 784, 2 x 392, 4 x 196, 7 x 112, 8 x 98, 14 x 56, 16 x 49, or 28 x 28
• 784 is a perfect square. √784 = 28

But that’s not the only thing remarkable about this perfect square: √784 is 28, the 7th triangular number, so like all other squared triangular numbers 784 has this additional property:

Just as 784 is a perfect square, five of the twelve clues in today’s puzzle are also perfect squares. But don’t let that fact trick you into writing the same factor in both the first column and the top row every time!

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 784:

784 can be written as the sum of three squares in exactly one way:

• 24² + 12² + 8² = 784

784 is also a palindrome in Bases 13, 17, and 27:

• 484 BASE 13; note that 4(169) + 8(13) + 4(1) = 784
• 2C2 BASE 17 (C is 12 base 10); note that 2(289) + 12(17) + 2(1) = 784
• 121 BASE 27; note that 1(27²) + 2(27) + 1(1) = 784

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