# Puzzle ZOCW

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My local greengrocer is a would-be mathematician.

She likes to arrange the apples in nice rows.

When she lays the apples in rows of 3, she has one left over.

And, when she lays them in rows of 5, she also has one left over.

Remarkably, she also has one left over when she arranges them in rows of 7 and 9.

But 11 seems to be the magic number, because in rows of 11 there are no apples left over.

How many apples does she have?

946 apples.

Reasoning
We know that (answer - 1) is divisible by 3, 5, 7, and 9.

If it's divisible by 9 then it's automatically divisible by 3.

So the number must be a multiple of 5, 7, and 9.

[5 x 7 x 9 = 315]

Since these numbers share no other common factors, we know that the answer must be a multiple of 315 (+1).

Let's try 1 x 315 + 1: 316 ÷ 11 = 28.73 - this doesn't work.

Let's try 2 x 315 + 1: 631 ÷ 11 = 57.36 - this doesn't work.

Let's try 3 x 315 + 1: 946 ÷ 11 = 86 - this works and is the answer.

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