One of the great early Beatles tracks. With ALG 17, we're standing on the number line and seeing how our position is related to multiples, especially multiples of 2 and 3.
MONDAY: Where do multiples of 3 occur on the number line? How can we tell whether a multiple of 3 is also a multiple of 6? And of 9??
Part b) can be solved by simply going through the multiples of 7 and 8 in the standard times tables, which leads us to 63, 64. However, a key feature here is the relation of these numbers to 56 (= 7×8) .... Or one can make use of 'the difference of two squares': 8×8 is a multiple of 8, 8×8 – 1 = (8–1)(8+1) is a multiple of 7. [This generalises very nicely, eg to finding consecutive numbers that are multiples of 19 and 20 respectively.]
THURSDAY: A classic 'think of a number ....' task. The task has links with Tuesday's version of ALG 17 and one can get quite a long way by using an empirical, data-generating approach. However, its attraction lies in the fact that one can use some fairly routine algebra to throw light on the underlying structure. [This is all the more noteworthy, given that we often use algebra to ease our path to an answer while ignoring structure!]
FRIDAY: We take an explicit look at the structure of Thursday's task.