Problem 10 · 2020 Math Kangaroo
Medium
Number Theory
place-valuecasework
In the addition on the right, different letters stand for different digits. Assuming the sum is correct, what is the greatest possible value of \(C + A + N\)?
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Answer: D — 21
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Hint 1 of 2
Read the columns of CAN + GUR = UUU with their carries; the result is a three-digit repdigit UUU.
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Hint 2 of 2
To make C + A + N large, push the units and the carried digits, allowing a leading zero in the second number.
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Approach: analyse CAN + GUR = UUU column by column, then maximise C + A + N
- Looking at the tens column, A plus the units carry must reach a multiple of 10, which forces A = 9 with a carry of 1.
- Trying the largest digits for C and N that still close the columns gives 498 + 057 = 555 (here G = 0 is allowed).
- That makes C + A + N = 4 + 9 + 8 = 21, option D, and no arrangement beats it among the choices.
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