- 725 is a composite number.
- Prime factorization: 725 = 5 x 5 x 29, which can be written 725 = (5^2) x 29
- 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 725 has exactly 6 factors.
- Factors of 725: 1, 5, 25, 29, 145, 725
- Factor pairs: 725 = 1 x 725, 5 x 145, or 25 x 29
- Taking the factor pair with the largest square number factor, we get √725 = (√25)(√29) = 5√29 ≈ 26.925824.

Give this Level 2 puzzle a try!

Print the puzzles or type the solution on this excel file: 12 Factors 2015-12-28

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Perhaps, you would like to know some other facts about the number 725:

725 can be expressed as the sum of consecutive numbers five different ways:

- 362 + 363 = 725; that’s
**2**consecutive numbers. - 143 + 144 +
**145**+ 146 + 147 = 725; that’s**5**consecutive numbers. - 68 + 69 + 70 + 71 + 72 + 73 + 74 + 75 + 76 + 77 = 725; that’s
**10**consecutive numbers. - 17 + 18 + 19 + . . . +
**29**+ . . . + 39 + 40 + 41 = 725; that’s**25**consecutive numbers. - 11 + 12 + 13 + . . . +
**25**+ . . . + 37 + 38 + 39 = 725; that’s**29**consecutive numbers.

725 is also the sum of the eleven prime numbers from 43 to 89.

The factors in one of its factor pairs, 25 x 29, are both 2 numbers away from their average, 27, so 725 is just 4 numbers away from perfect square 27² = 729 . Thus, 25 x 29 = (27 – 2)(27 + 2) = 27² – 2² = 729 – 4 = 725.

725 is the sum of two squares three different ways:

- 26² + 7² = 725
- 25² + 10² = 725
- 23² + 14² = 725

Because ALL of its prime factors have a remainder of one when divided by four, 725 is the hypotenuse of primitive Pythagorean triples:

- 364-627-725 which was calculated using 2(26)(7), 26² – 7², 26² + 7²
- 333-644-725 which was calculated using 23² – 14², 2(23)(14), 23² + 14²

It is also the hypotenuse of FIVE other Pythagorean triples.

- 85-720-725
- 120-715-725
- 203-696-725
- 435-580-725
- 500-525-725

725 is a palindrome in two bases:

- 505 BASE 12; note that 5(144) + 0(12) + 5(1) = 725.
- PP BASE 28 (P = 25 base 10); note that 25(28) + 25(1) = 725.

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