# Seeing Powers of Five in Powers of Two and Vice Versa

The decimal representations of oppositely signed powers of two and powers of five look alike, as seen in these examples: 2-3 = 0.125 and 53 = 125; 5-5 = 0.00032 and 25 = 32. The significant digits in each pair of powers is the same, even though one is a fraction and one is an integer. In other words, a negative power of one base looks like a positive power of the other.

Powers of Two and Powers of Five that Look Alike

This relationship is not coincidence; it’s a by-product of how fractions are represented as decimals. I’ll show you simple algebra that proves it, as well as algebra that proves similar properties — in products involving negative powers.

# Cycle Length of Powers of Two Mod Powers of Ten

In my article “Patterns in the Last Digits of the Positive Powers of Two” I noted that the positive powers of two modulo 10m cycle with period 4·5m-1, starting at 2m. For example, the powers of two mod 10 cycle with period four: 2, 4, 8, 6, 2, 4, 8, 6, … . In this article, I’ll present my proof, which has two parts:

• Part 1 shows that the powers of two mod 5m cycle with period 4·5m-1, starting at 20.
• Part 2 shows that the powers of two mod 10m cycle with the same period as the powers of two mod 5m, starting at 2m.