# Factors and multiples activity

This activity must be presented after Prime factorization and LCM.

If possible, find a number that's 1 more than a multiple of 7, and 1 more than a multiple of 4.

If possible, find a number that's 3 more than a multiple of 5, and 8 more than a multiple of 10.

If possible, find a number that's 4 more than a multiple of 6, and 2 more than a multiple of 4.

We know that

- When 59 is divided by 5, the remainder is 4
- When 59 is divided by 4, the remainder is 3
- When 59 is divided by 3, the remainder is 2
- When 59 is divided by 2, the remainder is 1

Find the smallest number \(n\) such that

- When \(n\) is divided by 10, the remainder is 9
- When \(n\) is divided by 9, the remainder is 8
- \(\ldots\)
- When \(n\) is divided by 2, the remainder is 1

We could rewrite the problem as

- \(n\) is one less than a multiple of 10.
- \(n\) is one less than a multiple of 9.
- \(\ldots\)
- \(n\) is one less than a multiple of 2.

The smallest number that satisfies these is

$$\begin{align} & \text{lcm}(2, 3, 4, 5, 6, 7, 8, 9, 10) - 1 \\ = {} & \text{lcm}\left(2, 3, 2^2, 5, 2 \cdot 3, 7, 2^3, 3^2, 2 \cdot 5\right) - 1 \\ = {} & 2^3 \cdot 3^2 \cdot 5 \cdot 7 - 1 \\ = {} & 2519 \end{align}$$