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.

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In my article “Patterns in the Last Digits of the Positive Powers of Two” I noted that the positive powers of two modulo 10^m cycle with period 4·5^m-1, starting at 2^m. 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 5^m cycle with period 4·5^m-1, starting at 2⁰.
• Part 2 shows that the powers of two mod 10^m cycle with the same period as the powers of two mod 5^m, starting at 2^m.

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