Example One: As far as I can tell, the sum of the digits of any product with a factor of 9 will always be 9.
Will always be divisible by 9 itself, or will sum to 9 if you keep summing the digits. (So 999 sums to 27, which sums to 9, so the original number is divisible by 9, as are the intermediate steps.) Very useful for accountants, because a transposition error is always divisible by 9, so if I'm off by $27, it means I probably transposed digits.
Related: A number divisible by 3 has digits that sum to a number divisible by 3.
Example Two: Similiar effect for products with a factor of 6, except that the sum of the digits is always either 3, 6 or 9.
Side-effect of the general "divisible by 3" rule. Anything divisible by 6 is by definition also divisible by 3.
I was just wondering if there is an explanation for it. If not, writing this should help me stop pondering it.
Yes, and my recollection is that the explanations are pretty simple, particularly for the "change in the digits" ones.
