- 6 is the first perfect number because 1 + 2 + 3 = 6. Also notice that (2^1)(2^2 – 1) = 2 x
**3**= 6. - 28 is the second perfect number because 1 + 2 + 4 + 7 + 14 = 28. Hmm… (2^2)(2^3 – 1) = 4 x
**7**= 28. - (2^3)(2^4 – 1) = 8 x
**15**= 120 is NOT a perfect number because 1 + 2 + 3 + 4 + 5 + 6 + 8 + 10 + 12 + 15 + 20 + 24 + 30 + 40 + 60 = 240 which is two times what it needs to be. - 496 is (2^4)(2^5 – 1) = 16 x
**31**. So why is 496 the third perfect number? Everything you need to know to figure out the answer to that question can be found somewhere in this post. -
2016 is (2^5)(2^6 – 1) = 32 x
**63**, and 2016 is also NOT a perfect number.

6, 28, and 496 are all triangular numbers as well as hexagonal numbers, but 120 and 2016 can also make that claim.

The clues in yesterday’s Find the Factors puzzle were all perfect squares. Today’s puzzle is only a little more difficult. You can solve it, too!

Print the puzzles or type the solution on this excel file: 12 Factors 2015-05-18

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- 496 is a composite number.
- Prime factorization: 496 = 2 x 2 x 2 x 2 x 31, which can be written 496 = (2^4) x 31
- The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 x 2 = 10. Therefore 496 has exactly 10 factors.
- Factors of 496: 1, 2, 4, 8, 16, 31, 62, 124, 248, 496
- Factor pairs: 496 = 1 x 496, 2 x 248, 4 x 124, 8 x 62, or 16 x 31
- Taking the factor pair with the largest square number factor, we get √496 = (√16)(√31) = 4√31 ≈ 22.271057

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Perfect numbers are another great mystery in number theory…