Using Euclid's Division Algorithm, find the largest number that divides 1251, 9377, and 15628 leaving remainders 1, 2, and 3 respectively.

The largest number that divides 1251, 9377, and 15628 leaving remainders 1, 2, and 3 respectively is 625. To find this number, subtract the given remainders from each number and calculate the HCF using Euclid’s Division Algorithm.

Given numbers after subtracting remainders:

1251 − 1 = 1250
9377 − 2 = 9375
15628 − 3 = 15625

Now find the HCF of 1250, 9375, and 15625.

Using Euclid’s Division Algorithm:

9375 = 1250 × 7 + 625
1250 = 625 × 2 + 0

So, HCF of 9375 and 1250 = 625

Now,

15625 = 625 × 25 + 0

Therefore, the HCF of all three numbers is 625. Hence, 625 is the largest number that divides the given numbers and leaves the required remainders.