
Shortcut Trick
The greatest number leaving the same remainder is the HCF of the differences of the given numbers.
Differences: (61 − 31) = 30, (91 − 61) = 30, (91 − 31) = 60.
x = HCF(30, 30, 60) = 30.
Required LCM = LCM(30, 48).
Since 30 = 6 × 5 and 48 = 6 × 8, LCM = 6 × 5 × 8 = 240.
∴ The correct answer is 240.

Alternate Method
Given:
The numbers are 31, 61, and 91.
x is the greatest number that leaves the same remainder when dividing these numbers.
Formula Used:
If a number leaves the same remainder when dividing a, b, and c, then the greatest such number is the HCF of |a − b|, |b − c|, and |c − a|.
Calculation:
⇒ First difference = 61 − 31 = 30
⇒ Second difference = 91 − 61 = 30
⇒ Third difference = 91 − 31 = 60
⇒ x = HCF of 30, 30, and 60
⇒ x = 30
⇒ We need to find the LCM of x and 48, which is the LCM of 30 and 48.
⇒ Prime factorization of 30 = 2 × 3 × 5
⇒ Prime factorization of 48 = 24 × 3
⇒ LCM(30, 48) = highest power of each prime factor present
⇒ LCM(30, 48) = 24 × 3 × 5 = 16 × 15 = 240
∴ The correct answer is 240.

Additional Information
Greatest Number Dividing with Same Remainder
When asked for the greatest number that divides a, b, and c leaving the same remainder, compute the HCF of their absolute differences: |a − b|, |b − c|, and |c − a|.
Least Common Multiple (LCM)
The smallest positive integer that is exactly divisible by each of the given numbers. It is found by multiplying the highest power of all prime factors involved.
Highest Common Factor (HCF)
The largest positive integer that divides two or more numbers without leaving a remainder. It is the product of the lowest power of each common prime factor.