Monday, January 1, 2007

Math is Fun - The Answer

The question was: Find the smallest positive integer (base 10), composed entirely of the digits 0 and 1 that is an integer multiple of 225.


First notice that 225 = 9 * 25 so the correct solution will be a multiple of both 9 and 25.

Starting with the factor 25...any multiple of 25 will end with one of 25, 50, 75, or 00. Obviously 00 is the only potential ending that meets the criteria of only having the digit 0 or 1.

Next, we know from this post that the sum of all the digits of any multiple of 9 is also divisible by 9. Since we are restricted to just using ones and zeros, the smallest number divisible by 9 will be 9 consecutive ones.

Combining those previous two observations we get the solution that the smallest positive integer that is evenly divisible by 225 is: 11,111,111,100.

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