Question

Yusuf has between 10 and 20 toffees. If he counts his toffees in 4s, he has 3 left over. If he counts his toffees in 5s, he has 4 left over. How many toffees has Yusuf got?

Answer 1

19 toffees.

Explanation:

It is given that Yusuf has between 10 and 20 toffees.

Let n be the number of toffees.

If he counts his toffees in 4s, he has 3 left over.

`n\equiv 3(mod4)` ... (1)

So, the possible values of n between 10 and 20 are 11, 15 and 19.

If he counts his toffees in 5s, he has 4 left over.

`n\equiv 4(mod5)` ... (2)

So, the possible values of n between 10 and 20 are 14 and 19.

From (1) and (2) the possible value of n between 10 and 20 is 19.

Therefore, Yusuf has 19 toffees.

Answer 2

Yusuf got 19 toffees

Explanation:

The problem tells us that if Yusuf counts his toffees in 4's, he has 3 left. Let's call the number of toffees "t", then we know that
`t`≡
`3(mod4)`

We also know that
`t`≡
`4(mod 5)`

Between 10 and 20, there are only 2 numbers t≡
`4(mod 5)`, these numbers are 14 and 19.

However, 14≡
`2(mod 4)` so it doesn't add up.

On the other hand, 19 ≡3
`(mod4)` and therefore, this is the answer.