Divisible by 9
MJNL and I went to Twilight Tea with MIBAgentQ and BL.
During the meal, BL was mentioning some neat math tricks where if you take a number, switch two digits, and then take the difference between the new number and the original number, the difference is divisible by 9.
I told to her how it could be explained by simple algebra, with a fairly generic argument that the operation was the same as taking the difference of a digit multiplied by two different whole powers of 10. The difference would be the original digit multiplied by a number that was always divisible by 9, since the difference between any two whole powers of 10 is always a multiple of 9. For example: 100 - 1 = 99, or 10 - 1000 = -990.
I thought about it for a bit and realized that the difference of any two integers will always be divisible by 9 so long as the signs of the two numbers match (both positive or both negative) and that both numbers use the same digits (order does not matter).
I think I'll post a proof later. If I do, I'll also repost my proofs for the Divisible Digits Number, and Number of Numbers that disappeared from the internet when my server died. (There are still links to those proofs so I better put something up.)
During the meal, BL was mentioning some neat math tricks where if you take a number, switch two digits, and then take the difference between the new number and the original number, the difference is divisible by 9.
I told to her how it could be explained by simple algebra, with a fairly generic argument that the operation was the same as taking the difference of a digit multiplied by two different whole powers of 10. The difference would be the original digit multiplied by a number that was always divisible by 9, since the difference between any two whole powers of 10 is always a multiple of 9. For example: 100 - 1 = 99, or 10 - 1000 = -990.
I thought about it for a bit and realized that the difference of any two integers will always be divisible by 9 so long as the signs of the two numbers match (both positive or both negative) and that both numbers use the same digits (order does not matter).
I think I'll post a proof later. If I do, I'll also repost my proofs for the Divisible Digits Number, and Number of Numbers that disappeared from the internet when my server died. (There are still links to those proofs so I better put something up.)
I've finally got around to posting a proof.
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