## A Multiplication Based Logic Puzzle

### 763 and Level 3

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. Stetson.edu informs us 763 is the smallest number whose 4th power can make that claim.

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