#math trivia for #June18: #170 can be factored three different ways into the form (a+b)*(c+d) where a, b, c, d are squares. What are they?

— Burt Kaliski Jr. (@modulomathy) June 19, 2012

Start by listing the possible factorizations of 170:

- 1*170
- 2*85
- 5*34
- 10*17

Among the eight factors, the ones that can be expressed as sums of squares are:

- 2 = 1 + 1
- 5 = 4 + 1
- 10 = 9 + 1
- 17 = 16 + 1
- 34 = 25 + 9
- 85 = 81 + 4
- 170 = 169 + 1

This means that the last three factorizations all have the appropriate form:

170 = (1 + 1) * (81 + 4) = (4 + 1) * (25 + 9) = (9 + 1) * (16 + 1)

And I suppose, in retrospect, I should also have included 1 as a sum of squares, because 1 = 1 + 0. As a result, there’s fourth answer that works, 170 = (1 + 0) * (169 + 1). I must have been thinking of only positive squares.

An interesting question is why 170 has this special property about its factorizations. Fermat’s Theorem on the sums of two squares states that an odd prime can be expressed as the sum of two squares if and only if it is one more than a multiple of four. The odd prime factors of 170 — 5 and 17 — both have this form. But this doesn’t explain why the other factors of 170 also do. What is special about 170? A good question for further exploration.