Problem 22 · 2010 Math Kangaroo
Stretch
Number Theory
cryptarithmdivisibility
In the multiplication of a three-digit number by a one-digit number, PPQ × Q = RQ5Q, the letters P, Q and R stand for different digits. What is P + Q + R?
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Answer: D — 17
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Hint 1 of 2
Look at the last digit: \(Q \times Q\) must end in \(Q\) again, which narrows \(Q\) down to very few digits.
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Hint 2 of 2
Once you know \(Q\), try the few three-digit numbers \(PPQ\) until the product matches the pattern \(RQ5Q\).
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Approach: use the last digit, then test
- The last digit of \(PPQ \times Q\) is the last digit of \(Q \times Q\), and it must equal \(Q\); the only digit that works in a 4-digit product is \(Q = 6\) (since \(6 \times 6 = 36\) ends in 6).
- Testing \(PP6 \times 6\), the value \(776 \times 6 = 4656\) fits \(RQ5Q = 4\,6\,5\,6\), giving \(P = 7,\ Q = 6,\ R = 4\).
- So \(P + Q + R = 7 + 6 + 4 = 17\) — the answer is D.
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