**Solution:**

Since, 1, 2 and 3 are the remainders of 1251, 9377 and 15628 respectively.

So, 1251 – 1 = 1250 is exactly divisible by the required number,

9377 – 2 = 9375 is exactly divisible by the required number,

15628 – 3 = 15625 is exactly divisible by the required number.

So, required number = HCF of 1250, 9375 and 15625.

By Euclid’s division algorithm,

15625 = 9375 x 1 + 6250

9375 = 6250 x 1 + 3125

6250 = 3125 x 2 + 0

=> HCF (15625, 9375) = 3125

3125 = 1250 x 2 + 625

1250 = 625 x 2 + 0

HCF(3125, 1250) = 625

So, HCF (1250, 9375, 15625) = 625

**Hence, the largest number is 625.**