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77777.... (2017-04-08 03:07:20)
 分类： 数学

A curious puzzle from the Penguin Problems Book, 1940:

A certain number consisting entirely of 7s is divisible by 199. Find the last four digits of the quotient, without finding the entire quotient.

 199 × the quotient equals the number we’re seeking (7777 …). Hence (200 × the quotient) minus the quotient will give us the same number. So let a, b, c, and d be the last four digits of the quotient. Now the quotient multiplied by 200 ends in r, s, 0, 0, where r, s are the digits that result from multiplying c, d by 2. Here’s the subtraction: So d must be 3, and c must be 2. Therefore s, which we know is twice d, is 6, and r, which is twice c, is 4. And that tells us that b is 8 and a is 6. So the last four digits of the quotient are 6823.
from https://www.futilitycloset.com/2017/04/02/77777/

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