Problem 24 · 2009 AMC 8
Hard
Algebra & Patterns
cryptarithm
The letters A, B, C, and D represent digits. If AB + CA = DA and AB − CA = A, what digit does D represent?
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Answer: E — 9.
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Hint 1 of 2
Attack the UNITS column first — it's almost never tangled by carries. In the addition, the units give B + A ending in A, which forces B = 0.
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Hint 2 of 2
Once B = 0 you know AB = 10A (a round number). Feed that into the subtraction AB − CA = A to pin down A and C, then read off D.
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Approach: crack the units column, then use the subtraction
- Units of the addition: B + A ends in A. Adding B leaves the last digit unchanged only if B = 0 (and there's no carry). So B = 0, making AB = 10A.
- Subtraction: 10A − (10C + A) = A ⇒ 9A − 10C = A ⇒ 8A = 10C ⇒ 4A = 5C.
- 4A = 5C needs A divisible by 5 and C by 4 (single digits): A = 5, C = 4.
- Then DA = AB + CA = 50 + 45 = 95, so D = 9 (and the units '5' matches A — consistent).
- Why this transfers: in any cryptarithm, start where carries are simplest (units of a sum) to lock one letter, then substitute forward. A relation like 4A = 5C is solved by divisibility, not by trying all 100 pairs.
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