• 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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• 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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• 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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• 718 is a composite number.
• Prime factorization: 718 = 2 x 359
• 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 718 has exactly 4 factors.
• Factors of 718: 1, 2, 359, 718
• Factor pairs: 718 = 1 x 718 or 2 x 359
• 718 has no square factors that allow its square root to be simplified. √718 ≈ 26.795522.

Today’s puzzle looks like a Christmas Star.

Print the puzzles or type the solution on this excel file: 10 Factors 2015-12-21

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Here’s a few facts about the number 718:

178 + 179 + 180 + 181 = 718 making it the sum of 4 consecutive numbers.

718 is the sum of three squares three different ways:

• 3² + 15² + 22² = 718
• 9² + 14² + 21² = 718
• 13² + 15² + 18² = 718

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• 711 is a composite number.
• Prime factorization: 711 = 3 x 3 x 79, which can be written 711 = (3^2) x 79
• 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 711 has exactly 6 factors.
• Factors of 711: 1, 3, 9, 79, 237, 711
• Factor pairs: 711 = 1 x 711, 3 x 237, or 9 x 79
• Taking the factor pair with the largest square number factor, we get √711 = (√9)(√79) = 3√79 ≈ 26.66458.

Here is today’s puzzle:

Print the puzzles or type the solution on this excel file: 12 Factors 2015-12-14

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Even though in English 711 is a rather fun rhyming number, it has been difficult for me to find interesting facts about it:

7 + 1 + 1 = 9 so 711 is divisible by both 3 and 9.

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

• 355 + 356 = 711; that’s two consecutive numbers.
• 236 + 237 + 238 = 711; that’s three consecutive numbers.
• 75 + 76 + 77 + 78 + 79 + 80 + 81 + 82 + 83 = 711; that’s nine consecutive numbers.

Is 711 a palindrome in any other base? Not in any base less than 78, but in BASE 78 it is palindrome 99 because 9(78) + 9(1) = 711.

We could ALMOST write that 711 is 11 7 in BASE 64 because 11(64) + 7(1) = 711. However, we can’t REALLY write that because base 10’s eleven would most likely be represented by the letter B in base 64. Thus we would likely write 711 as B7 in Base 64.

Since the words “eleven” and “twelve” do not contain any reference to the word “ten”, I suppose in English we could say that 711 is eleven 7 in base 64. We likely wouldn’t be able to make that same claim in other languages, however.

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