## 8. Bag of Apples

When asked how many apples are in the bag, the seller replied: "If I count them in twos, or in threes, or in fours, or in fives, or in sixes, there is always one left over. If I count them in sevens, none are left over." Determine the smallest number of apples that could be in the bag.

## Result: $\quad 301$
## Solution.

The number of apples in the bag can be evenly divided by 7, and when divided by 2, 3, 4, 5, and 6, the remainder is 1. This means that the number of apples minus 1 can be divided (without a remainder) by 2, 3, 4, 5, and 6.

If it can be evenly divided by 4, it can also be divided by 2.

If it can be evenly divided by 4 and 3, it can also be divided by 12.

If it can be evenly divided by 12, it can also be divided by 6.

If it can be evenly divided by 12 and 5, it can also be divided by 60.

These could be the numbers $60, 120, 180, 240, 300, 360, 420, 480, \ldots$

The desired number is 1 greater and must be able to be evenly divided by 7.

$61: 7 = 8$ with a remainder of 5.

$121: 7 = 17$ with a remainder of 2.

$181: 7 = 25$ with a remainder of 6.

$241: 7 = 34$ with a remainder of 3.

$301: 7 = 43$.

The smallest number of apples that can be in the bag is 301.