# 1472 and Level 4

### Today’s Puzzle:

Can you use logic to figure out where the factors from 1 to 12 must go to make the given clues be the products of those factors?

### Factors of 1472:

• 1472 is a composite number.
• Prime factorization: 1472 = 2 × 2 × 2 × 2 × 2 × 2 × 23, which can be written 1472 = 2⁶ × 23.
• 1472 has at least one exponent greater than 1 in its prime factorization so √1472 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1472 = (√64)(√23) = 8√23
• The exponents in the prime factorization are 6 and 1. Adding one to each exponent and multiplying we get (6 + 1)(1 + 1) = 7 × 2 = 14. Therefore 1472 has exactly 14 factors.
• The factors of 1472 are outlined with their factor pair partners in the graphic below.

### A Little More about the Number 1472:

1472 is the difference of two squares in FIVE different ways:
369² – 367² = 1472
186² – 182² = 1472
96² – 88² = 1472
54² – 38² = 1472
39² – 7² = 1472