Problem of the Week 1117

Binary Divisibility

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If one thinks of the binary representation of a number as a base-ten number, we can ask: Which numbers are factors of their binary selves?

For instance, 2 is a factor of 10, 10 is a factor of 1010, and 20 is a factor of 10100. If a number has this property (binary divisibility) and I multiply it by 2 or 10, the new product has that property as well.

Aside from powers of 2 and 10, what other numbers have this property? Find at least two.

Is there a pattern we can use to generate some or all of them?

© Copyright 2009 Dave Ehren and Shilad Sen. Reproduced with permission.