What is the largest number which divides 77, 147 and 252 to leave the same…

Concept

When one number divides several numbers and leaves the SAME remainder each time, that remainder cancels out if we subtract one number from another. So the divisor divides every pairwise DIFFERENCE exactly. The largest such divisor is therefore the HCF (greatest common divisor) of the differences.

Trick (one line): largest number leaving the same remainder = HCF of the differences of the numbers.

Why it works

If N leaves remainder r for two numbers a and b, then a = Nq1 + r and b = Nq2 + r. Subtracting, b - a = N(q2 - q1), which is a multiple of N. So N divides (b - a). Hence N divides every difference, and the biggest possible N is the HCF of those differences.

Application

1. Take the differences of the given numbers: 147 - 77 = 70, and 252 - 147 = 105. (252 - 77 = 175 is automatically covered, since 175 = 70 + 105.)

2. Find the HCF of these differences. Prime factorise: 70 = 2 × 5 × 7 and 105 = 3 × 5 × 7.

3. Common prime factors are 5 and 7, so HCF = 5 × 7 = 35.

Therefore the largest number is 35.

Cross-check

Divide each original number by 35: 77 = 35 × 2 + 7, 147 = 35 × 4 + 7, 252 = 35 × 7 + 7. The remainder is 7 in every case, confirming 35 works.