612 is the hypotenuse of the Pythagorean triple 288-540-612. Which factor of 612 is the greatest common factor of those three numbers?

612 = 17 x 36, which is 17 x 18 x 2, and that is exactly four times the formula of the 17^{th} triangular number. Thus . . .

612 = 4(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17)

612 can be expressed as the sum of consecutive counting numbers in several ways:

- 612 = 203 +
**204**+ 205 (**3**consecutive numbers because it is divisible by 3) - 612 = 73 + 74 + 75 + 76 + 77 + 78 + 79 + 80 (eight consecutive numbers because it is divisible by 4, but not 8)
- 612 = 64 + 65 + 66 + 67 +
**68**+ 69 + 70 + 71 + 72 (**9**consecutive numbers because it is divisible by 9) - 612 = 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 +
**36**+ 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 (**17**consecutive numbers because it is divisible by 17) - 612 = 14 + 15 + 16 + 17 + . . . . + 34 + 35 + 36 + 37 (24 consecutive numbers because 612 is divisible by 12, but not by 24)

What is a relationship between the numbers in bold print and the number 612?

612 is also the sum of the twelve **prime** numbers from 29 to 73.

Print the puzzles or type the solution on this excel file: 12 Factors 2015-09-07

—————————————————————————————————

- 612 is a composite number.
- Prime factorization: 612 = 2 x 2 x 3 x 3 x 17, which can be written 612 = (2^2) x (3^2) x 17
- The exponents in the prime factorization are 2, 2 and 1. Adding one to each and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 x 3 x 2 = 18. Therefore 612 has exactly 18 factors.
- Factors of 612: 1, 2, 3, 4, 6, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204, 306, 612
- Factor pairs: 612 = 1 x 612, 2 x 306, 3 x 204, 4 x 153, 6 x 102, 9 x 68, 12 x 51, 17 x 36 or 18 x 34
- Taking the factor pair with the largest square number factor, we get √612 = (√36)(√17) = 6√17 ≈ 24.73863

Although I prefer using a modified cake method to find square roots, most people prefer factor trees. If you use a factor tree, I suggest you still look for easy-to-detect perfect square factors (100, 4, 9, 25) so that the most common duplicate prime factors are together:

—————————————————————————————————