Why must 2 digits of any 2-digit number that is multiple of 9 (eg 27) add up to 9?
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People are talking about addition, which is true, but this is actually easier by subtraction.
9 is 10 – 1. Additionally, 99 is 100 – 1, 999 is 1000 – 1, and so on… or “any power of ten is always one more than a multiple of nine.”
What a number actually means (ie 268) is: 2×10^2 + 6×10^1 + 8×10^0. This is what we call *base 10* because every digit represents an order of magnitude of 10. This is why zeros don’t matter.
Anyway, back to the question. 268 is two 100s (99+99+2), six 10s (9+9+9+9+9+9+6), eight 1s. Notice how the remainders pile up. If they sum to any multiple of nine, then the number must be divisible by nine. Try writing it out with something like 20412.
The tricky part is that nine only has this special property in base 10. In base 5 (where twelve is 22), the magic number is 4. Always one less than the base (or a divisor of the magic number, like 3 in base 10). The key here is the infinite -1 multiples at the top of this comment.
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