In the alphametic equation SEND + MORE = MONEY, where each letter represents a unique...
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In the alphametic equation SEND + MORE = MONEY, where each letter represents a unique distinct digit from 0 to 9, what is the numerical value of letter M?
Show answer & explanation
In the addition of two 4-digit numbers resulting in a 5-digit number, the carry-over to the ten-thousands place can only be 1, as the maximum possible sum of two 4-digit numbers (including carry) is 9999 + 9999 + 1 = 19999.
Step-by-step Derivation:
Step 1: Analyze the structure of the equation: SEND (4 digits) + MORE (4 digits) = MONEY (5 digits).
Step 2: Let the carry from the thousands column (S + M) to the ten-thousands column be denoted as C4. Since we are adding two digits (S and M) and potentially a carry from the hundreds column (C3), the maximum value of S + M + C3 is 9 + 8 + 1 = 18.
Step 3: The ten-thousands digit of the result (MONEY) is M. Therefore, M = C4.
Step 4: In any addition of two n-digit numbers, the carry to the (n+1)-th position can only be 0 or 1. Since M is the leading digit of a 5-digit number, M cannot be 0.
Step 5: Therefore, M must be 1.
Step 6: (Verification) If M = 1, then S + 1 (+ carry C3) = 10 + O. Since S is at most 9, S + 1 + 1 = 11, meaning O must be 0 or 1. Since M=1 and digits are unique, O must be 0. This is consistent with the known solution to this classic puzzle (9567 + 1085 = 10652).