Summing ABC

Number Theory Level 2

A B C A B + A 9 8 7 \begin{array} { l l l l l } & A & B & C \\ & & A & B \\ + & & & A \\ \hline & 9 & 8 & 7 \\ \end{array}

In the above cryptogram, all the letters represent distinct (non-negative) digits, and the leading digit of a number cannot be 0.

What is A B C ‾ \overline{ABC} ?


The answer is 890.

Chung Kevin by Chung Kevin , 45, Australia

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2 solutions

Marta Reece
May 4, 2017

Starting with the hundreds place: A has to be an 8 or a 9. But if it were a 9, then no carry would be allowed, and in tens place we would have a problem: A + B = 9 + B could not give 8. Therefore A = 8, and there is 1 carried to the hundreds place.

Tens place: To get 8 in tens place plus the 1 carried into hundreds place, we need 18 from 8 + B + possible carry. That can only be done with B = 9 and 1 carried from units place. So we have B = 9.

Units place: We have A + B = 8 + 9 = 17 without the C. The 1 will carry over, the 7 satisfies the equation. So C = 0.

Number A B C ‾ = 890 \overline{ABC}=\boxed{890}

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. .
Mar 19, 2021

If you see this, you will understand.

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