Question

If you add to a number and then multiply by 12, the product is 324. What was the original number?

Answer 1

It's because 9 is one less than the base of the number system we use. The same property holds for 6 in base 7 arithmetic for example. Allow me to attempt to demonstrate an inductive argument.

It's because 9 is one less than the base of the number system we use. The same property holds for 6 in base 7 arithmetic for example. Allow me to attempt to demonstrate an inductive argument.The base case is 9, whose digits add up to 9, a multiple of 9. Now suppose x is a multiple of 9 whose digits also add up to a multiple of 9, and consider x+9. If the last digit of x is 0, it becomes 9 and the digit sum increases by 9 (and remains a multiple of 9). Otherwise, we decrease the last digit by 1 and add 1 to the second last digit, not changing the overall digit sum. The only case not yet accounted for is if the second last digit is 9, we reset it to 0 (changing the digit sum by a factor of 9) and carry the 1 to the next digit, continuing this process until we reach a non-nine digit. Either way, the 1 subtracted from the last digit ends up being added somewhere, and all digit resets in between change the sum by 9.

Hopefully that wasn't too contrived for you to follow

Answer 2

Answer:

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