# 1517 and Level 5

### Today’s Puzzle:

Which common factor of 72 and 36 is needed to solve this puzzle? Is it 6, 9, or 12? There is an easier place to begin this level 5 puzzle. Don’t guess and check. Use logic to know which factors you should use.  You can figure it out!

### Factors of 1517:

• 1517 is a composite number.
• Prime factorization: 1517 = 37 × 41.
• 1517 has no exponents greater than 1 in its prime factorization, so √1517 cannot be simplified.
• The exponents in the prime factorization are 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1517 has exactly 4 factors.
• The factors of 1517 are outlined with their factor pair partners in the graphic below.

### More Facts about the Number 1517:

1517 is the difference of two squares in two different ways:
759² – 758² = 1517,
39² – 2² = 1517.

1517 is also the sum of two squares in two different ways:
34² + 19² = 1517,
29² + 26² = 1517.

1517 is the hypotenuse of FOUR Pythagorean triples:
165-1508-1517, calculated from 29² – 26², 2(29)(26), 29² + 26²,
333-1480-1517, which is 37 times (9-40-41),
492-1435-1517, which is (12-35-37) times 41,
795-1292-1517, calculated from 34² – 19², 2(34)(19), 34² + 19².