### Today’s Puzzle:

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-29

Here is a logical order to use the clues to solve the puzzle:

### Factors of 540:

- 540 is a composite number.
- Prime factorization: 540 = 2 x 2 x 3 x 3 x 3 x 5, which can be written 540 = (2^2) x (3^3) x 5
- The exponents in the prime factorization are 2, 3 and 1. Adding one to each and multiplying we get (2 + 1)(3 + 1)(1 + 1) = 3 x 4 x 2 = 24. Therefore 540 has exactly 24 factors.
- Factors of 540: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, 270, 540
- Factor pairs: 540 = 1 x 540, 2 x 270, 3 x 180, 4 x 135, 5 x 108, 6 x 90, 9 x 60, 10 x 54, 12 x 45, 15 x 36, 18 x 30 or 20 x 27
- Taking the factor pair with the largest square number factor, we get √540 = (√36)(√15) = 6√15 ≈ 23.237900077

### Sum-Difference Puzzles:

60 has six factor pairs. One of those pairs adds up to 17, and another one subtracts to 17. Put the factors in the appropriate boxes in the first puzzle.

540 has twelve factor pairs. One of the factor pairs adds up to 51, and a different one subtracts to 51. If you can identify those factor pairs, then you can solve the second puzzle!

The second puzzle is really just the first puzzle in disguise. Why would I say that?

### More about the Number 540:

540 is the sum of the fourteen consecutive prime numbers from 13 to 67. Can you list all those prime numbers? It is also the sum of consecutive prime numbers 269 and 271.

540 has the same number of factors as 504. Both of those numbers tie with 360, 420, and 480 for the most factors so far.

540 is the hypotenuse of the Pythagorean triple 324-432-540. What is the greatest common factor of those three numbers?

540 is also an untouchable number.

The sum of the interior angles of every convex pentagon total 540 degrees.

Ah! Another untouchable number so soon?!?! What a wonderful thought to ponder. 🙂

Yes! Ponder away! There are infinitely many of these untouchable numbers, and they can come up so unexpectedly!