• 754 is a composite number.
• Prime factorization: 754 = 2 x 13 x 29
• 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 754 has exactly 8 factors.
• Factors of 754: 1, 2, 13, 26, 29, 58, 377, 754
• Factor pairs: 754 = 1 x 754, 2 x 377, 13 x 58, or 26 x 29
• 754 has no square factors that allow its square root to be simplified. √754 ≈ 27.459060.

Here’s today’s puzzle:

Print the puzzles or type the solution on this excel file: 12 Factors 2016-01-25

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I have much more to tell you about the number 754:

754 is the sum of consecutive numbers three different ways:

• 187 + 188 + 189 + 190 = 754; that’s 4 consecutive numbers.
• 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 + 61 + 62 + 63 + 64 = 754; that’s 13 consecutive numbers.
• 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 = 754; that’s 29 consecutive numbers.

Because all of the odd prime factors of 754 can be expressed as the sum of two squares, 754 can also be expressed as the sum of two squares:

• 27² + 5² = 754
• 23² + 15² = 754

Also because two of its prime factors are hypotenuses of primitive Pythagorean triples, I knew that 754 is the hypotenuse of FOUR Pythagorean triples:

• 270² + 704² = 754²; the triple 270-704-754 was calculated from 2(27)(5), 27² – 5², 27² + 5².
• 290² + 696² = 754²
• 304² + 690² = 754²; the triple 304-690-754 was calculated from 23² – 15², 2(23)(15), 23² + 15².
• 520² + 546² = 754²

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

• 27² + 4² + 3² = 754
• 24² + 13² + 3² = 754
• 23² + 12² + 9² = 754
• 21² + 13² + 12² = 754

754 is a palindrome in three bases:

• 626 BASE 11; note that 6(121) + 2(11) + 6(1) = 754.
• 2F2 BASE 16 (F is 15 base 10); note that 2(256) + 15(16) + 2(1) = 754.
• QQ BASE 28(Q is 26 base 10); note that 26(28) + 26(1) = 754.

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• 752 is a composite number.
• Prime factorization: 752 = 2 x 2 x 2 x 2 x 47, which can be written 752 = (2^4) x 47
• The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 x 2 = 10. Therefore 752 has exactly 10 factors.
• Factors of 752: 1, 2, 4, 8, 16, 47, 94, 188, 376, 752
• Factor pairs: 752 = 1 x 752, 2 x 376, 4 x 188, 8 x 94, or 16 x 47
• Taking the factor pair with the largest square number factor, we get √752 = (√16)(√47) = 4√47 ≈ 27.422618.

Here’s today’s puzzle. A logical way to solve it is given in the table at the end of the post.

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

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Here’s a few more thoughts about the number 752:

52 is divisible by 4 so 752 is also divisible by 4. However, 52 is not also divisible by 8, but since 7 is odd, 752 IS divisible by 8.

8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 = 752; that’s 32 consecutive numbers.

752 is the sum of two consecutive primes: 373 + 379 = 752.

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Today’s Puzzle:

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

Some possible steps to solve the puzzle:

Factors of 750:

• 750 is a composite number.
• Prime factorization: 750 = 2 x 3 x 5 x 5 x 5, which can be written 750 = 2 x 3 x (5^3)
• The exponents in the prime factorization are 1, 1, and 3. Adding one to each and multiplying we get (1 + 1)(1 + 1)(3 + 1) = 2 x 2 x 4 = 16. Therefore 750 has exactly 16 factors.
• Factors of 750: 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 125, 150, 250, 375, 750
• Factor pairs: 750 = 1 x 750, 2 x 375, 3 x 250, 5 x 150, 6 x 125, 10 x 75, 15 x 50, or 25 x 30
• Taking the factor pair with the largest square number factor, we get √750 = (√25)(√30) = 5√30 ≈ 27.386127875.

Sum-Difference Puzzles:

30 has four factor pairs. One of those pairs adds up to 13, and another one subtracts to 13. Put the factors in the appropriate boxes in the first puzzle.

750 has eight factor pairs. One of the factor pairs adds up to ­65, and a different one subtracts to 65. If you can identify those factor pairs, then you can solve the second puzzle!

The second puzzle is really just the first puzzle in disguise. Why would I say that?

More reasons to be interested in the number 750:

750 can be written as the sum of consecutive numbers seven ways:

• 249 + 250 + 251 = 750; that’s 3 consecutive numbers.
• 186 + 187 + 188 + 189 = 750; that’s 4 consecutive numbers.
• 148 + 149 + 150 + 151 + 152 = 750; that’s 5 consecutive numbers.
• 57 + 58 + 59 + 60 + 61 + 62 + 63 + 64 + 65 + 66 + 67 + 68 = 750; that’s 12 consecutive numbers.
• 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57 = 750; that’s 15 consecutive numbers.
• 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 = 750; that’s 20 consecutive numbers.
• 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 = 750; that’s 25 consecutive numbers.

750 is also the sum of all the prime numbers from 17 to 79. Do you remember what those sixteen prime numbers are?

Because 5, 25, and 125 are its factors, 750 is the hypotenuse of three Pythagorean triple triangles:

• 210² + 720² = 750²
• 264² + 702² = 750²
• 450² + 600² = 750²

750² = 562500 which is another cool looking square number whose digits include 5 and the value of 5^4.

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

• 26² + 7² + 5² = 750
• 25² + 11² + 2² = 750
• 25² + 10² + 5² = 750
• 23² + 14² + 5² = 750
• 23² + 11² + 10² = 750
• 19² + 17² + 10² = 750

Wikipedia tells us that 750 is the 15th nonagonal number because 15(7⋅15 – 5)/2 = 750. It is also 10 times the 5th nonogonal number because 10 ⋅ 5(7⋅5 – 5)/2 = 750.

750 is a palindrome in three different bases:

• 23232 BASE 4; note that 2(4^4) + 3(4^3) + 2(4^2) + 3(4^1) + 2(4^0) = 750.
• 2A2 BASE 17 (A= 10 base 10); note that 2(17²) + 10(17) + 2(1) = 750.
• PP BASE 29 (P = 25 base 10); note that 25(29) + 25(1) = 750.

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.

• 748 is a composite number.
• Prime factorization: 748 = 2 x 2 x 11 x 17, which can be written 748 = (2^2) x 11 x 17
• 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 748 has exactly 12 factors.
• Factors of 748: 1, 2, 4, 11, 17, 22, 34, 44, 68, 187, 374, 748
• Factor pairs: 748 = 1 x 748, 2 x 374, 4 x 187, 11 x 68, 17 x 44, or 22 x 34
• Taking the factor pair with the largest square number factor, we get √748 = (√4)(√187) = 2√187 ≈ 27.34958866.

Here is today’s puzzle:

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

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Here is some more about composite number 748:

748 can be written as the sum of consecutive numbers 3 ways:

• 90 + 91 + 92 + 93 + 94 + 95 + 96 + 97 = 748; that’s 8 consecutive numbers.
• 63 + 64 + 65 + 66 + 67 + 68 + 69 + 70 + 71 + 72 + 73 = 748; that’s 11 consecutive numbers.
• 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 = 748; that’s 17 consecutive numbers.

Because 17 is one of its factors, 748 is the hypotenuse of Pythagorean triple 352-660-748, and 352² + 660² = 748².

748 is the sum of 3 squares two different ways. Both ways contain a duplicate square.

• 26² + 6² + 6² = 748
• 18² + 18² + 10² = 748

748 is palindrome MM in BASE 33 (M = 22 base 10); note that 22(33) + 22(1) = 748.

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• 747 is a composite number.
• Prime factorization: 747 = 3 x 3 x 83, which can be written 747 = (3^2) x 83
• The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2  = 6. Therefore 747 has exactly 6 factors.
• Factors of 747: 1, 3, 9, 83, 249, 747
• Factor pairs: 747 = 1 x 747, 3 x 249, or 9 x 83
• Taking the factor pair with the largest square number factor, we get √747 = (√9)(√83) = 3√83 ≈ 27.3313007.

It’s Steve Morris’s birthday so I thought I’d make him a cake, but no regular size cake will do. He has been one of my earliest supporters, and I know that sometimes even a level 6 puzzle is just too easy for him. Once he sent out this tweet:

This maths puzzle took me 6 minutes – can you beat that? http://t.co/8bX7yjFKRt via @IvaSallay

— Steve Morris (@SteveMorris999) June 1, 2014

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Admittedly that puzzle was easier than most level 6’s, but recently he tweeted me a puzzle that I have had to start over more than once and still haven’t conquered:

@findthefactors @NerdintheBrain @abyssbrain Can you solve GCHQ’s Christmas card puzzle? (GCHQ is Britain’s NSA) https://t.co/YrKXBeFRMq

— Steve Morris (@SteveMorris999) December 10, 2015

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I think Steve Morris is due for an extra difficult Find the Factors puzzle for his birthday, one that all the numbers from 1 to 16 can be the factors. I’ll wait at least a week before I give any hints to complete it, too. As always, there is only one solution, but it can be found using logic.

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

The possible factors for each clue is given below. Adding 14, 15, and 16 as possible factors really complicates the puzzle!

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Now I’ll write a little about the number 747:

747 is a palindrome in base 10. Boeing’s most recognizable airplane also bares that number.

747 can be written as the sum of consecutive numbers five different ways:

• 373 + 374 = 747; that’s 2 consecutive numbers
• 248 + 249 + 250 = 747; that’s 3 consecutive numbers
• 122 + 123 + 124 + 125 + 126 + 127 = 747; that’s 6 consecutive numbers
• 79 + 80 + 81 + 82 + 83 + 84 + 85 + 86 + 87 = 747; that’s 9 consecutive numbers
• 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 = 747; that’s 18 consecutive numbers

747 is the sum of three squares six different ways. Three of those ways repeat squares.

• 27² + 3² + 3² = 747
• 25² + 11² + 1² = 747
• 23² + 13² + 7² = 747
• 21² + 15² + 9² = 747
• 19² + 19² + 5² = 747
• 17² + 17² + 13² = 747

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Since today’s puzzle is the biggest puzzle I have ever published, it could be a little difficult just noticing that a factor had been duplicated in the top row or first column. Here is the tweet Steve Morris sent out once he finally solved the puzzle:

@findthefactors pic.twitter.com/VwFKb50GoW

— Steve Morris (@SteveMorris999) January 24, 2016

//platform.twitter.com/widgets.js

Now after waiting over a week, I now reveal one of the ways to solve this difficult puzzle logically:

• 746 is a composite number.
• Prime factorization: 746 = 2 x 373
• 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 746 has exactly 4 factors.
• Factors of 746: 1, 2, 373, 746
• Factor pairs: 746 = 1 x 746 or 2 x 373
• 746 has no square factors that allow its square root to be simplified. √746 ≈ 27.313000567.

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 746:

746 is the sum of 4 consecutive numbers: 185 + 186 + 187 + 188 = 746.

746 is the hypotenuse of a Pythagorean triple triangle because 504² + 550² = 746².

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

•  27² + 4² + 1² = 746
• 24² + 13² + 1² = 746
• 24² + 11² + 7² = 746
• 21² + 17² + 4² = 746
• 21² + 16² + 7² = 746
• 20² + 15² + 11² = 746

OEIS.org shared this fun number fact: 1^7 + 2^4 + 3^6 = 746.

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