Solution to: Odd Oranges

Assume the number of oranges is a. Then a - 1 is divisible by 3, 5, 7, and 9. Therefore, a - 1 is a multiple of 5 × 7 × 9 = 315 (note: 9 is also a multiple of 3, so 3 should not be included!). We are looking for a value of n for which 315 × n + 1 is divisible by 11. After some trial and error, it turns out that the smallest n for which this holds is n = 3. This means that the greengrocer has at least 946 oranges.

Note that for n = 14, 25, 36, etc. (each time adding 11), it also holds that 315 × n + 1 is divisible by 11.

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