In my article “Patterns in the Last Digits of the Positive Powers of Five” I noted that the positive powers of five modulo 10^m cycle with period 2^m-2, m ≥ 2, starting at 5^m. In this article, I’ll present my proof, which has two parts:

• Part 1 shows that the powers of five mod 2^m cycle with period 2^m-2, m ≥ 2, starting at 5⁰.
• Part 2 shows that the powers of five mod 10^m cycle with the same period as the powers of five mod 2^m, starting at 5^m.

The highlight of my proof is in part 1, where I derive a formula to show that the period, or order, of 5 mod 2^m is 2^m-2. While it is in general not possible to derive a formula for the order of a number, I’ll show it is possible for the powers of five mod 2^m — due to a hidden, binary structure I’ve uncovered.

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People have been tweeting about the upcoming dates that look like binary numbers. 10/10/10 seems to be a favorite, both because of its symmetry and because 101010 = 42 in decimal (you know, the answer to the ultimate question of life, the universe, and everything). Here are the nine dates in 2010, interpreted as binary numbers, and with their decimal equivalents:

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The positive powers of five — 5, 25, 125, 625, 3125, 15625, … — have a compact, repeating pattern in their ending m digits, in the powers of five from 5^m on. For example: starting with 5, their last digit is always 5; starting with 25, their last two digits are always 25; starting with 125, their last three digits alternate between 125 and 625. These cycles come in lengths of powers of two.

Cycles in the Ending Digits of the Powers of Five

I will show you why these cycles exist, how they are expressed mathematically, and how to visualize them.

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