Shortcut Trick
Find the LCM of the given divisors: LCM(2, 3, 4) = 12.
The required number will be a multiple of 12 plus the remainder 1: (12 × k) + 1.
The smallest 3-digit multiple of 12 is 108 (since 12 × 9 = 108).
Add the remainder to this multiple: 108 + 1 = 109.
∴ The correct answer is 109.
Alternate Method
Given:
Divisors = 2, 3, and 4
Remainder = 1 in each case
Formula Used:
Number = LCM(divisors) × k + Remainder
Calculations:
⇒ LCM of 2, 3, and 4 = 12
⇒ The required number must be of the form 12k + 1
⇒ We need the smallest 3-digit natural number, so 12k + 1 ≥ 100
⇒ 12k ≥ 99
⇒ k ≥ 8.25
⇒ Since k must be an integer, the smallest possible value for k is 9
⇒ Required number = 12 × 9 + 1
⇒ Required number = 108 + 1 = 109
∴ The correct answer is 109.
Additional Information
LCM Remainder Concept (Same Remainder)
When a number is divided by x, y, and z and leaves the same remainder r in each case, the general form of the number is: LCM(x, y, z) × k + r
LCM Remainder Concept (Different Remainders)
When a number divided by x, y, and z leaves remainders a, b, and c respectively, and (x − a) = (y − b) = (z − c) = d, the number is of the form: LCM(x, y, z) × k − d
HCF and LCM Relation
For any two numbers A and B, the product of the numbers is equal to the product of their HCF and LCM: A × B = HCF × LCM