Of which five natural numbers is the sum equal to the product?
If a, b, c, d and e are the five natural numbers ordered by ascending size, then abcde = a + b + c + d + e ≤ 5e. This means that abcd ≤ 5. We can now distinguish between five cases.
The first case is abcd = 1, which means a = b = c = d = 1. In that case, e = 4 + e and is impossible.
The second case is abcd = 2, which means a + b + c + d = 1 + 1 + 1 + 2. In that case, 2e = 5 + e and thus e = 5. This gives the solution (1, 1, 1, 2, 5).
The third case is abcd = 3, which means a + b + c + d = 1 + 1 + 1 + 3. In that case, 3e = 6 + e and thus e = 3. This gives the solution (1, 1, 1, 3, 3).
The fourth case is abcd = 4, which means either a + b + c + d = 1 + 1 + 2 + 2 = 6 or a + b + c + d = 1 + 1 + 1 + 4 = 7. Thus, 4e = 6 + e or 4e = 7 + e. The first equation suggests e = 2 and the solution (1, 1, 2, 2, 2). The second equation does not give an integer e.
The last case is abcd = 5, which means a + b + c + d = 1 + 1 + 1 + 5 = 8. In that case, 5e = 8 + e or e = 2. It gives no additional solution.
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This puzzle originally appeared in Spektrum der Wissenschaft and was reproduced with permission.