5A) B=price of 1 biscuit, M=price of carton of milk:
2B+M=5, 3B+M=6.50, so B=6.50-5=$1.50, therefore M=5-3=$2. M-B=$0.50 or 50c more than a biscuit.
5B) 27/N has a remainder of 3 is another way of saying qN+3=27, where q is an integer, qN=24. The factors of 24 bigger than 3 but less than 24 are 4,6,8,12. So there are 4 possible integers for N.
5C) We need only consider the third and fourth layers because, in the other three layers the cubes will be exposed and so painted on at least one face. In the third layer, the central cube is completely hidden, and in the fourth layer 4 cubes are hidden. Therefore 5 cubes will have no painted faces.
5D) 1591=1600-9=(40+3)(40-3)=43×37, the sum of these prime factors is 43+37=80.
5E) The cups can be represented by letters (initial letters of the colours) which represent numbers in relationships. "To the left" means "less than", and "to the right" means "greater than".
So G<R, Y<P, and there are two conditions for B: either (1) B<G, P, R, Y; or (2) B>G, P, R, Y.
Let's look at (1):
B<G<R and B<Y<P, so we know B is the smallest. There are 24 arrangements for the remaining letters but only 6 of these satisfying the other conditions: BGRYP, BGYPR, BGYRP, BYGPR, BYGRP, BYPGR.
For (2) G<R<B and Y<P<B, we therefore have GRYPB, GYPRB, GYRPB, YGPRB, YGRPB, YPGRB, where B is the largest.
So in all there are 12 arrangements.