A Question of Quotients

Dec 17, 2016

RAY: Pick out a three digit number and write it down. Now, repeat those three digits. So, if you picked 2-7-1, you now have 271271. Now I want you to divide that number by seven.

Now, if you have a remainder, put it off to the side. Like if the remainder's one, two, whatever it is, put it off to the side. Take the quotient that you got and divide that by 11. And whatever remainder you have, put that off to the side too.

TOM: Got it, yeah.

RAY: Now, divide that remaining quotient, the quotient you just got, by thirteen. Another prime number! OK, now take that number that you've got, and the quotients, the remainders that you had from the previous divisions, and add them all together.

TOM: Add what together?

RAY: Add the remainders and the last quotient that you got. And you're going to wind up with the original number. So, the question is, why does this work?

Answer: 

RAY: When you take a number like 1, 2, 3, or 4, 5, 6, or 7, 2, 1, and multiply it by a thousand and one, you wind up with the same number repeated. So if you start with 4, 5, 6, and multiply that by 1,001, you get 4, 5, 6, 4, 5, 6, don't you? And then all you're doing now is dividing it by the factors of 1,001. Which happen to be, some of which happen to be, 7, 11, and 13. So I knew you weren't going to come out with any remainders.

TOM: You little devil, you.

RAY: And that's the reason it works, no matter what the three numbers are. Do we have a winner?

TOM: We do have a winner. Congratulations to John Slater from Wheaton, Illinois!


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