What is the largest number which divided 62, 132, 237 to leave same remainder…

Key idea: if a number leaves the same remainder when dividing several numbers, it divides all pairwise differences of those numbers.

• Compute the relevant differences: 132 - 62 = 70.

• Compute the other difference: 237 - 62 = 175.

• Therefore the required divisor must divide both 70 and 175.

Find the greatest common divisor of 70 and 175. Using the Euclidean algorithm: 175 = 70×2 + 35, and 70 = 35×2 + 0, so gcd(70, 175) = 35.

Thus the largest number that divides 62, 132, and 237 leaving the same remainder is 35.

Quick verification: 62 = 35×1 + 27, 132 = 35×3 + 27, 237 = 35×6 + 27, so the common remainder is 27.

Answer: 35