Q. Find the greatest number which divides 99, 123 and 183 leaving the same remainder in each case.

Explanation by: Vaibhav Shukla

Let t be the remainder. Then (99 – t), (123 – t) and (183 – t) will be exactly divisible by the required number. As discussed under division method of HCF, any number which divides the given number, also divides their difference. In other words, HCF of given numbers is same as the HCF of their difference.
∴ Required number = HCF of (123 – t) – (99 – t), (183 – t) – (123 – t) and (183 – t) – (99 – t)
Required number = HCF of (123 – 99), (183 – 123) and (183 – 99)
Required number = HCF of 24, 60 and 84
Now, 24 = 2 × 2 × 2 × 3
60 = 2 × 2 × 3 × 5
84 = 2 × 2 × 3 × 7
∴ Required HCF = 2 × 2 × 3 = 12
∴ Required number = 13