Using Euclid’s division algorithm, find the largest number that divides 1251, 9377 and 15628 leaving remainders 1, 2 and 3, respectively.
625
Step 1: Understand the problem.
We want a number that divides each of the three given numbers, but leaves a remainder: 1 when dividing 1251, 2 when dividing 9377, and 3 when dividing 15628.
Step 2: Subtract the remainders.
If a number leaves remainder \(r\), then it divides the number minus \(r\).
So, the required number must divide:
\(1251 - 1 = 1250\)
\(9377 - 2 = 9375\)
\(15628 - 3 = 15625\)
Step 3: Find the HCF of these adjusted numbers.
We now need the HCF (highest common factor) of 1250, 9375, and 15625.
Step 4: Apply Euclid’s division algorithm.
First, find HCF(9375, 1250).
Divide 9375 by 1250:
\(9375 = 1250 \times 7 + 625\)
Now divide 1250 by 625:
\(1250 = 625 \times 2 + 0\)
So HCF(9375, 1250) = 625.
Step 5: Include the third number.
Now check HCF(15625, 625).
Divide 15625 by 625:
\(15625 = 625 \times 25 + 0\)
So the HCF is 625.
Final Answer: The largest required number is 625.