In the following equation, each of the letters represents a uniquely different digit in base ten. Suppose:
(YE)(ME) = TTT, where the number YE is less than ME in the product on the left.
Then the sum E + M + T + Y equals:
- 19
- 20
- 21
- 22
- 24
| If YE and ME each end in |
then TTT ends in |
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| 1 | 1 | ← | No good (T ≠ E). | |||||||||||||||
| 2 | 4 | ← | 444 = 4 · 111 = 4 · 3 · 37. But 37 is prime. And 12 x 37 does not work (2 ≠ 7). |
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| 3 | 9 | ← | 999 = 9 · 3 · 37 = 27 · 37 = 999. But 7 ≠ 3. | |||||||||||||||
| 4 | 6 | ← | 666 = 6 · 3 · 37 = 18 · 37 = 666, no good. | |||||||||||||||
| 5 | 5 | ← | 555 = 5 · 3 · 37 = 15 · 37 = 555, no good. | |||||||||||||||
| 6 | 6 | ← | No good, see above. | |||||||||||||||
| 7 | 9 | ← |
This works! We have
And 2 + 7 + 3 + 9 = 21. |
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| 8 | 4 | ← | No good, see above. | |||||||||||||||
| 9 | 1 | ← | 111 = 3 · 37, impossible. | |||||||||||||||
| 0 | 0 | ← | No good (T ≠ E). |
So 27 · 37 is the only product that works, and the answer is (c).