2019/20 AHSMC Part 1 Problem 11

Problem 11 of 16

$A$ and $B$ are positive integers such that (i) $A + B = 2019$ , (ii) all the digits of $A$ are odd and all the digits of $B$ are even, and (iii) $A$ and $B$ are as close together as possible with these properties. The largest prime number dividing into $B$ is

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Solution

For $A$ and $B$ to be closest together, the first digit of the larger number must be $1$ (rather than $2$ ), so $A > B$ . Thus the largest possible value for $B$ is $888$ , which works because it means that $A = 1131$ , which does have all odd digits. Since $888 = 2^3 \cdot 3 \cdot 37$ , the answer is $37$ . The answer is (E).

University of Alberta

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