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| Math Central | Quandaries & Queries |
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Question from Ben, a teacher: Question: Using mathematical induction, prove that if the sum of the digits of a number is divisible by three, then the number itself is also divisible by 3. |
Ben,
You need to think about what a number like 9826 means.
9823 = 9(10^3) + 8(10^2) + 2(10) + 6 = 9(999+1) + 8(99+1) + 2(9+1) + 6 = 9(999) + 8(99) + 2(9) + 9 + 8 + 2 + 6,
now clearly all the terms with all the '9's' in them are a multiple of 3 so the original number is a multiple of 3 if and only if 9 + 8 + 2 + 6 is a multiple of 3.
To use induction you would induct on the number of digits - at the induction step you would pull off the 1st digit and it accompanying power of 10 and apply the induction hypothesis to the rest.
Penny
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