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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• 776 is a composite number.
• Prime factorization: 776 = 2 x 2 x 2 x 97, which can be written 776 = (2^3) x 97
• 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 776 has exactly 8 factors.
• Factors of 776: 1, 2, 4, 8, 97, 194, 388, 776
• Factor pairs: 776 = 1 x 776, 2 x 388, 4 x 194, or 8 x 97
• Taking the factor pair with the largest square number factor, we get √776 = (√4)(√194) = 2√194 ≈ 27.85677655

Try solving today’s 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 776:

776 is the sum of two squares: 26² + 10² =776.

776 is the hypotenuse of Pythagorean triple 520-576-776 so 520² + 576² = 776².

776 is also the sum of three squares five different ways:

• 26² + 8² + 6² = 776
• 24² + 14² + 2² = 776
• 24² + 10² + 10² = 776
• 22² + 16² + 6² = 776
• 18² + 16² + 14² = 776

776 is a palindrome in three other bases:

• 646 BASE 11; note that 6(121) + 4(11) + 6(1) = 776
• 272 BASE 18; note that 2(18²) + 7(18) + 2(1) = 776
• 161 BASE 25; note that 1(25²) + 6(25) + 1(1) = 776

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• 770 is a composite number.
• Prime factorization: 770 = 2 x 5 x 7 x 11
• 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 770 has exactly 16 factors.
• Factors of 770: 1, 2, 5, 7, 10, 11, 14, 22, 35, 55, 70, 77, 110, 154, 385, 770
• Factor pairs: 770 = 1 x 770, 2 x 385, 5 x 154, 7 x 110, 10 x 77, 11 x 70, 14 x 55, or 22 x 35
• 770 has no square factors that allow its square root to be simplified. √770 ≈ 27.74887.

Here is a puzzle for you to solve:

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

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Here is more information about the number 770:

Because 5 is one of its factors, 770 is the hypotenuse of a Pythagorean triple, and 462² + 616² = 770².

770 is the sum of the squares of three consecutive numbers: 15² + 16² + 17² = 770.

770 can also be written as the sum of three squares seven other ways:

• 27² + 5² + 4² = 770
• 25² + 12² + 1² = 770
• 25² + 9² + 8² = 770
• 24² + 13² + 5² = 770
• 23² + 15² + 4² = 770
• 20² + 19² + 3² = 770
• 20² + 17² + 9² = 770

770 is palindrome MM in Base 34 (M = 22 base 10); note that 22(34) + 22(1) = 770.

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It’s obvious that 763 is divisible by 7 so it is a composite number.

• 763 is a composite number.
• Prime factorization: 763 = 7 x 109
• 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 763 has exactly 4 factors.
• Factors of 763: 1, 7, 109, 763
• Factor pairs: 763 = 1 x 763 or 7 x 109
• 763 has no square factors that allow its square root to be simplified. √763 ≈ 27.6224546.

Now try solving today’s puzzle:

Print the puzzles or type the solution on this excel file: 10 Factors 2016-02-04

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Here’s a little more about the number 763:

763 is the sum of consecutive numbers three different ways:

• 381 + 382 = 763; that’s 2 consecutive numbers.
• 106 + 107 + 108 + 109 + 110 + 111 + 112 = 763; that’s 7 consecutive numbers.
• 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 + 61 = 763; that’s 14 consecutive numbers.

763 is also the sum of consecutive prime numbers two different ways:

• 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 + 103 = 763; that’s 9 consecutive primes.
• 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 = 763; that’s 17 consecutive primes.

763 is the hypotenuse of a Pythagorean triple, and 420² + 637² = 763².

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

• 27² + 5² + 3² = 763
• 23² + 15² + 3² = 763

763^4 = 338,920,744,561, a number in which every digit appears at least one time. OEIS.org informs us 763 is the smallest number whose 4th power can make that claim.

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Obviously 749 can be evenly divided by 7.

• 749 is a composite number.
• Prime factorization: 749 = 7 x 107
• 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 749 has exactly 4 factors.
• Factors of 749: 1, 7, 107, 749
• Factor pairs: 749 = 1 x 749 or 7 x 107
• 749 has no square factors that allow its square root to be simplified. √749 ≈ 27.367864.

Here’s today’s puzzle:

Print the puzzles or type the solution on this excel file: 10 Factors 2016-01-18

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Here’s a little more about the number 749:

749 can be written as the sum of consecutive numbers three different ways:

• 374 + 375 = 749; that’s 2 consecutive numbers.
• 104 + 105 + 106 + 107 + 108 + 109 + 110 = 749; that’s 7 consecutive numbers.
• 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 = 749; that’s 14 consecutive numbers.

749 is the sum of three consecutive prime numbers: 241 + 251 + 257 = 749.

749 is the sum of three cubes eight different ways:

1. 27² + 4² + 2² = 749
2. 26² + 8² + 3² = 749
3. 24² + 13² + 2² = 749
4. 22² + 16² + 3² = 749
5. 22² + 12² + 11² = 749
6. 20² + 18² + 5² = 749
7. 19² + 18² + 8² = 749
8. 18² + 16² + 13² = 749

749 is a palindrome in two different bases:

• 525 BASE 12; note that 5(144) + 2(12) + 5(1) = 749
• 1C1 BASE 22 (C = 12 base 10); note that 1(22²) + 12(22) + 1(1) = 749

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A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. (None of the factors are greater than 10.)  Write the corresponding factors in the factor column (1st column) and factor row (top row).  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.