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Question Consider the numbers 72 and 144. Suppose they are factorised into composite numbers as: 72 = 6 × 12 and 144 = 8 × 18. Seeing this, can one say that these two numbers have no common factor other than 1? Why not?
No, one cannot say that 72 and 144 have no common factor other than 1 because their factorisations have composite numbers. Here 72 = 6 × 12 144 = 8 × 18 These are not prime factorisations. Both 6 and 12 are composite numbers, as are 8 and 18. Prime factorisation of 72 = 2 × 2 × 2 × 3 × 3 Prime factorisation of 144 = 2 × 2 × 2 × 2 × 3 × 3 ∴ HCF(72, 144) = 2 × 2 × 2 × 3 × 3 = 8 × 9 = 72 Since the HCF is 72, which is greater than 1, the numbers have common factors other than 1. 3.2 Least, but not Last! Figure It Out (Page 58)