5. The positive integer 1 and every fourth integer after that are coloured red; 2 and every fourth integer after that are coloured blue; 3 and every fourth integer are coloured yellow and 4 and every fourth integer are coloured green. Peter picks a number coloured yellow and a number coloured blue and adds them together. What could the colour of his answer be?
A blue or green
B only green
C only yellow
D only blue
E only red
Solution
E

When Peter picks a number coloured yellow and a number coloured blue, he picks a number that is 3 more than a multiple of 4 and a number that is 2 more than a multiple of 4 . Therefore, when he adds his two numbers, he obtains a number that is $2+3=5$ more than a multiple of 4 . Since 5 is itself 1 more than a multiple of 4 , this means his number is 1 more than a multiple of 4 and hence is coloured red.