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Algebra Difficulty 4.9 AIME Prove it United States

Problem:

Consider the addition problem:

	C	A	S	H
+			M	E
O	S	I	D	E

where each letter represents a base-ten digit, and

C, M, O \neq 0

. (Distinct letters are allowed to represent the same digit) How many ways are there to assign values to the letters so that the addition problem is true?

Solution

Solution:

Clearly, $C A S H$ and $M E$ cannot add up to $11000$ or more, so $O=1$ and $S=0$ . By examining the units digit, we find that $H=0$ . Then $C A S H+M E<9900+99<10000$ , so there are no solutions.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.