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# Find the largest number which on dividing 1251, 9377 and 15628 leave remainders 1, 2 and 3 respectively.

Given: 1251, 9377 and 15628.

To find: Here we have to find the value of the largest number which divides 1251, 9377 and 15628 and leaves remainders of 1, 2 and 3 respectively.

Solution:

If the required number divide 1251, 9377 and 15628 leaving remainders 1, 2 and 3 respectively, then this means that number will divide 1250(1251 $-$ 1), 9375(9377 $-$ 2) and 15625(15628 $-$ 3) completely.

Now, we just have to find the HCF of 1250, 9375 and 15625.

First, let's find HCF of 1250 and 9375 using Euclid's division algorithm:

Using Euclid’s lemma to get:

- $9375\ =\ 1250\ \times\ 7\ +\ 625$

Now, consider the divisor 1250 and the remainder 625, and apply the division lemma to get:

- $1250\ =\ 625\ \times\ 2\ +\ 0$

The remainder has become zero, and we cannot proceed any further.

Therefore the HCF of 1250 and 9375 is the divisor at this stage, i.e., 625.

Now, let's find HCF of 625 and 15625 using Euclid's division algorithm:

Using Euclid’s lemma to get:

- $15625\ =\ 625\ \times\ 25\ +\ 0$

The remainder has become zero, and we cannot proceed any further.

Therefore the HCF of 625 and 15625 is the divisor at this stage, i.e., 625.

So, the largest number which divides 1251, 9377 and 15628 and leaves remainders of 1, 2 and 3 respectively is 625.

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