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The Base-Conversion Method: Why Does it Work?

Consider first the binary case:

When you divide a number by two, the remainder will be either a zero or a one. If the remainder is 0, then the base-ten number must have been even (that is, a multiple of two), so there will be no ones, and therefore the right-most digit will be "0". If the remainder is 1, then the base-ten number must have been odd (that is, one more than a multiple of two), so there will be a "1" as the right-most digit in the units column.   Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved

Now divide again by two. If the remainder is zero, then, after getting rid of any extra 1 (from being an odd number), the number that was left must be a multiple of four, so there won't be any 2 left over. Otherwise, there was a multiple of two left over, so there will be a 1 in the twos column.

Continue in like manner. Each time you divide, you're asking "Does the original number contain a multiple of this power of two?", and the remainder is either telling you "yes" (with a "0") or "no" (with a "1").

This reasoning carries through with other bases. If, say, you're converting to base eleven and you divide by 11, the remainder will tell you how many more than a multiple of eleven the given number is. Since this remainder value must be between 0 (if the number is an exact multiple of eleven) and 10 (one less than a multiple of eleven), then the digits in a base-eleven number will never contain one single solitary digit for "eleven".