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Because both of its prime factors are hypotenuses of primitive Pythagorean triples, 685 is the hypotenuse of FOUR Pythagorean triples. Two are primitive; two are not:
- 37-684-685 which was calculated from 19² – 18², 2(19)(18), 19² + 18²
- 156-667-685 which was calculated from 2(26)(3), 26² – 3², 26² + 3²
- 411-548-685 (What factor of 685 is the greatest common factor of those 3 numbers?)
- 440-525-685 (and what is their greatest common factor?)
As you may have notice from those calculations, 685 is the sum of two squares two different ways:
- 19² + 18² = 685
- 26² + 3² = 685
685 is the 19th centered square number because 18 and 19 are consecutive numbers and 19² + 18² = 685. There are 685 small squares of various colors in this graphic.
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- 685 is a composite number.
- Prime factorization: 685 = 5 x 137
- 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 685 has exactly 4 factors.
- Factors of 685: 1, 5, 137, 685
- Factor pairs: 685 = 1 x 685 or 5 x 137
- 685 has no square factors that allow its square root to be simplified. √685 ≈ 26.172505.
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