The required number divides (85 − 1) = 84 and (72 − 2) = 70.
So, the required number = HCF(84, 70).
$84 = 2^2 \times 3 \times 7$
$70 = 2 \times 5 \times 7$
HCF(84, 70) = $2 \times 7 = \mathbf{14}$
∴ The greatest required number is 14.
When a number divides 85 leaving remainder 1, it exactly divides 85 − 1 = 84. Similarly, it exactly divides 72 − 2 = 70. The greatest such number is the HCF of 84 and 70. Examiners expect you to show the subtraction step clearly, then find HCF by prime factorisation (or Euclid's division). Both steps carry marks.