The sum of the two 5 -digit numbers AMC10 and AMC12 is 123422. What is A + M + C?

2003 AMC 10A Problem 11

Attempt Problem 11 of the 2003 AMC 10A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2003 AMC 10A solutions, or check the answer key.

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11.

The sum of the two

5

-digit numbers

AMC10

and

AMC12

123422.

What is

A + M + C?

10

11

12

13

14

Write each number as

100 \cdot \overline{AMC}

plus its last two digits

Their sum is

200 \cdot \overline{AMC} + 22 = 123422

The two numbers equal

100 \cdot \overline{AMC} + 10

and

100 \cdot \overline{AMC} + 12,

so their sum is

200 \cdot \overline{AMC} + 22 = 123422.

Then

200 \cdot \overline{AMC} = 123400,

\overline{AMC} = 617.

Therefore

A + M + C = 6 + 1 + 7 = 14.

Thus, the correct answer is E.