2025/26 AHSMC Part 1 Problem 6

Problem 6 of 16

How many integers $N \ge 10$ have the property that removing the last digit of $N$ gives an integer strictly smaller than $\dfrac{N}{11}$ ?

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Solution

Write $N = 10M + d$ with last digit $d$ . The condition $M < N/11$ becomes $11M < 10M + d$ , i.e. $M < d$ . Thus $N$ is a two-digit number whose first digit is smaller than its second (and neither digit is $0$ ). Choosing two distinct digits from $\{1,\ldots,9\}$ and arranging them increasingly can be done in $\dbinom{9}{2} = 36$ ways. The answer is (E).

University of Alberta

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