# 1748 A Factor Fits Valentine’s Puzzle

Contents

### Today’s Puzzle:

Can you find the one and only solution to this Factor Fits Valentine puzzle or will it give you fits?

### Factors of 1748:

• 1748 is a composite number.
• Prime factorization: 1748 = 2 × 2 × 19 × 23, which can be written 1748 = 2² × 19 × 23.
• 1748 has at least one exponent greater than 1 in its prime factorization so √1748 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1748 = (√4)(√437) = 2√437.
• The exponents in the prime factorization are 2, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1748 has exactly 12 factors.
• The factors of 1748 are outlined with their factor pair partners in the graphic below.

### More About the Number 1748:

1748 is the difference of two squares in two ways:
438² – 436² = 1748, and
42² – 4² = 1748.
That means we are only 16 numbers away from the next perfect square!

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