The prompt was devised by Mark Richards, a maths teacher at Lancaster Girls' Grammar School in Lancaster (UK). He reports here on using it with a new year 10 class:
The prompt combines the potential for inductive exploration and deductive reasoning about the number of pairs that satisfy the conditions for the highest common factor and lowest common multiple. The four pairs of integers that satisfy the conditions in the prompt are 6 and 180, 12 and 90, 18 and 60, and 30 and 36. The inquiry could develop in different directions. Why, for example, are there no pairs for a HCF of 8 and a LCM of 140? When are there two pairs? Why is there never an odd number of pairs? Can there ever be more than four pairs?
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