Date: Tue, 27 Oct 1998 09:24:43 EST
To: QandQ@MathCentral.uregina.ca
Subject: help

We are trying to solve what SHOULD BE a simple 7th grade problem and have run into a wall.
It is:
ABCD
ABCD
ABCD
ABCD
-------
EBEA

It is an addition problem. Each particular letter represents the same digit. We are looking for what number "B" represents. We have tried a multitude of solutions and have come up empty.

Can you help.
Bruce

Hi Bruce
Notice that the sum of 4 A's plus whatever carry comes from the column of B's is a one digit number so A must be 1 or 2. Now look at the digits 0, 1, ..., 9 and the remainders when the sum of 4 of them is divided by 10.

 Digit 0 1 2 3 4 5 6 7 8 9 Remainder 0 4 8 2 6 0 4 8 2 6

The sum of 4 D's has a remainder of A when divided by 10 but and A is 1 or 2. But 1 does not appear in the remainder row in the table above so A must be 2.

Since A=2 the sum of four A's is 8 and E is a one digit number, the carry from the column of B's must be 0 or 1. Thus B cannot be larger than 4.

• B cannot be 4 as the sum of four 4's sum 16 and a carry of 8 would be required.

• B cannot be 2 as A is 2.

• B cannot be 1 as the sum of four 1's sum 4 and a carry of 7 would be required.

• If B=0 then E=8. An examination of the C's and D's columns shows that this choice for B is also impossible.

Hence B=3.

What remains now is to show that there is a choice for C and D that solves the puzzle.

Cheers,
Penny

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