### Today’s Puzzle:

Both 12 and 102 are factors of 1632. Something special happens when either one squares itself and looks in a mirror. Solving this puzzle from Math Happens will show you what happens to 12 and 12².

A Squaring @findthefactors puzzle for the @AustinChronicle thank you Iva Sallay! pic.twitter.com/gvhggeZuh7

— MathHappens (@MathHappensOrg) April 6, 2021

You can see that puzzle on page 33 of this e-edition or this pdf of the Austin Chronicle. You can find other Math Happens Puzzles here.

This next puzzle will help you discover what happens when 102 and 102² look in a mirror!

Why do you suppose the squares of (12, 21) and (102, 201) have that mirror-like property?

### Factor Trees for 1632:

There are many possible factor trees for 1632, but today I will focus on two trees that use factor pairs containing either 12 or 102:

### Factors of 1632:

- 1632 is a composite number.
- Prime factorization: 1632 = 2 × 2 × 2 × 2 × 2 × 3 × 17, which can be written 1632 = 2⁵ × 3 × 17.
- 1632 has at least one exponent greater than 1 in its prime factorization so √1632 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1632 = (√16)(√102) = 4√102.
- The exponents in the prime factorization are 5, 1, and 1. Adding one to each exponent and multiplying we get (5 + 1)(1 + 1)(1 + 1) = 6 × 2 × 2 = 24. Therefore 1632 has exactly 24 factors.
- The factors of 1632 are outlined with their factor pair partners in the graphic below.

### More about the Number 1632:

1632 is the hypotenuse of a Pythagorean triple:

768-1440-1632, which is (8-15-**17**) times **96**.

1632 is the difference of two squares in EIGHT different ways:

409² – 407² = 1632,

206² – 202² = 1632,

139² – 133² = 1632,

106² – 98² = 1632,

74² – 62² = 1632,

59² – 43² = 1632,

46² – 22² = 1632, and

41² – 7² = 1632.

That last difference of two squares means 1632 is only 49 numbers away from the next perfect square, 1681.