Correct Option: C

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