Question 14

Let $$A$$ be a 3-digit number with distinct nonzero digits and $$B$$ be the number obtained by reversing the digits of $$A$$. Determine the largest possible prime factor of $$|A - B|$$.

Correct Answer: 11

Solution

Writing $$A = 100a + 10b + c$$ and $$B = 100c + 10b + a$$ gives $$|A - B| = 99|a - c| = 3^2 \times 11 \times |a - c|$$. The digits are distinct and nonzero, so $$1 \le |a - c| \le 8$$ and every prime factor of $$|a - c|$$ is at most 7. The factor 11 always occurs and no larger prime can appear, so the largest possible prime factor is 11.

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