Q2. Luke has a set of cards numbered 1 to 8, which he arranges in pairs. He notes the sum of the numbers on each pair of cards and calculates the lowest common multiple (LCM) of the four sums. For example, the

pairs could have been

{7,3}, {2,6}, {5,1}, {8,4}.

In this case, the pair sums would have been 10, 8, 6, 12 respectively,

giving an LCM of 120. Pair sums are not necessarily different.

a) If Luke obtained an LCM of 24, find two different ways in which the

cards could have been paired.

b) Explain why Luke could not obtain an LCM of 39.

c) Luke now takes 20 cards numbered 1 to 20. Show how the cards can

be paired to give an LCM of 60.

d) Luke now takes six cards numbered 1 to 6 but he groups them into

two sets of three. He finds the sums of the numbers in each triple and

then finds the LCM of these two triple sums. What is the smallest

LCM he can obtain?