Anna is playing with some cars and divides them into three sets of equal size. However, when she tries to divide them into five sets of equal size, there are four left over. Given that she has fewer than 50 cars, what are the possible numbers of cars she can have?

METHOD 1

let x be the number of cars

we know \(x \equiv 0(\bmod 3)\)     (A1)

also \(x \equiv 4(\bmod 5)\)     (A1)

so \(x = 3t \Rightarrow 3t \equiv 4(\bmod 5)\)     M1

\( \Rightarrow 6t \equiv 8(\bmod 5)\)

\( \Rightarrow t \equiv 3(\bmod 5)\)

\( \Rightarrow t = 3 + 5s\)

\( \Rightarrow x = 9 + 15s\)     A1

since there must be fewer than 50 cars, x = 9, 24, 39     A1A1A1

Note: Only award two of the final three A1 marks if more than three solutions are given.

[7 marks] 

METHOD 2

x is a multiple of 3 that ends in 4 or 9     R4

therefore x = 9, 24, 39     A1A1A1     N3

Note: Only award two of the final three A1 marks if more than three solutions are given.

[7 marks]

There were a number of totally correct solutions to this question, but some students were unable to fully justify their results.