• 762 is a composite number.
• Prime factorization: 762 = 2 x 3 x 127
• 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 762 has exactly 8 factors.
• Factors of 762: 1, 2, 3, 6, 127, 254, 381, 762
• Factor pairs: 762 = 1 x 762, 2 x 381, 3 x 254, or 6 x 127
• 762 has no square factors that allow its square root to be simplified. √762 ≈ 27.604347.

This level 2 puzzle isn’t very difficult:

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 762:

People who memorize the digits of pi have to stop someplace. Wikipedia informs us that one place to stop is known as the Feynman point which is 999999 beginning at pi’s 762nd decimal place. It is named after Richard Feynman who reportedly said in a lecture that he would like to recite from memory the digits of pi up to that point because he could then end the recitation by saying 999999 and so on. Perhaps he would even be able to make pi sound like a rational number? Be sure to check out the highlighted digits of both pi and tau that appear in a graphic in that article. There is also an explanation of how truly unusual a sequence of six repeating digits can be.

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

• 253 + 254 + 255 = 762; that’s 3 consecutive numbers. (254 × 3 = 762)
• 189 + 190 + 191 + 192 = 762; that’s 4 consecutive numbers.
• 58 + 59 + 60 + 61 + 62 + 63 + 64 + 65 + 66 + 67 + 68 + 69  = 762; that’s 12 consecutive numbers.

762 can also be written as the sum of two consecutive prime numbers: 379 + 383 = 762, and as the sum of four consecutive prime numbers: 181 + 191 + 193 + 197 = 762.

762 is the sum of three squares three different ways:

• 25² + 11² + 4² = 762
• 23² + 13² + 8² = 762
• 20² + 19² + 1² = 762

762 is palindrome and repdigit 222 in BASE 19 because 2(19²) + 2(19) + 2(1) = 762.

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• 759 is a composite number.
• Prime factorization: 759 = 3 x 11 x 23
• 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 759 has exactly 8 factors.
• Factors of 759: 1, 3, 11, 23, 33, 69, 253, 759
• Factor pairs: 759 = 1 x 759, 3 x 253, 11 x 69, or 23 x 33
• 759 has no square factors that allow its square root to be simplified. √759 ≈ 27.5499546.

Here’s today’s puzzle. A logical way to find its solution is at the end of the post:

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

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What else can I say about the number 759?

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

• 379 + 380 = 759; that’s 2 consecutive numbers.
• 252 + 253 + 254 = 759; that’s 3 consecutive numbers.
• 124 + 125 + 126 + 127 + 128 + 129 = 759; that’s 6 consecutive numbers.
• 64 + 65 + 66 + 67 + 68 + 69 + 70 + 71 + 72 + 73 + 74 = 759; that’s 11 consecutive numbers.
• 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45  = 759; that’s 22 consecutive numbers.
• 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44  = 759; that’s 23 consecutive numbers.
• 7 + 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 = 759; that’s 33 consecutive numbers.

759 is also the sum of five consecutive prime numbers: 139 + 149 + 151 + 157 + 163 = 759.

759 is a palindrome in two bases:

• 3C3 BASE 14 (C is 12 base 10)
• NN BASE 32 (N is 23 base 10)

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• 755 is a composite number.
• Prime factorization: 755 = 5 x 151
• 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 755 has exactly 4 factors.
• Factors of 755: 1, 5, 151, 755
• Factor pairs: 755 = 1 x 755 or 5 x 151
• 755 has no square factors that allow its square root to be simplified. √755 ≈ 27.4772633.

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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Here’s a little more about 755:

755 is the sum of consecutive numbers three different ways:

• 377 + 378 = 755; that’s 2 consecutive numbers.
• 149 + 150 + 151 + 152 + 153 = 755; that’s 5 consecutive numbers.
• 71 + 72 + 73 + 74 + 75 + 76 + 77 + 78 + 79 + 80 = 755; that’s 10 consecutive numbers.

Because 5 is one of its factors, 755 is the hypotenuse of Pythagorean triple 453-604-755.

755 is the sum of three squares six different ways:

• 27² + 5² + 1² = 755
• 25² + 11² + 3² = 755
• 25² + 9² + 7² = 755
• 23² + 15² + 1² = 755
• 21² + 17² + 5² = 755
• 19² + 15² + 13² = 755

755 is palindrome 131 in BASE 26; note that 1(26²) + 3(26) + 1(1) = 755.

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• 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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