Articles from November, 2009

Seeing Powers of Five in Powers of Two and Vice Versa

By Rick Regan Contact e-mail icon (Published November 16th, 2009)

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

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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Cycle Length of Powers of Two Mod Powers of Ten

By Rick Regan Contact e-mail icon (Published November 6th, 2009)

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.

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