Question

There are two sections A and B of Grade X. There are 28 students in Section A and 30 students in Section B. What is the minimum number of books you will acquire for the class library so that they can be distributed equally among students of Section A or Section B ?

• A. 144
• B. 2
• C. 420
• D. 272

Hint:

Whenever a problem asks for the "minimum", "least", or "smallest" quantity that must satisfy multiple grouping conditions, it is almost always an L.C.M. problem.
Conversely, "maximum" or "largest" grouping problems usually require finding the H.C.F.

Answer 1

The Correct Option is C

Step 1: Restate what is actually being asked.
We need the smallest number of books that can be split evenly among 28 students and also split evenly among 30 students. Any such number must be a common multiple of 28 and 30, and the smallest one is the LCM.

Step 2: Find the HCF of 28 and 30 first, using repeated division.
Divide the bigger number by the smaller one and keep dividing the remainder into the previous divisor.
$30 = 1 \times 28 + 2$
$28 = 14 \times 2 + 0$
The remainder became 0, so the last non zero remainder, 2, is the HCF of 28 and 30.

Step 3: Use the product rule connecting HCF and LCM.
For any two numbers, the product of the HCF and the LCM equals the product of the numbers themselves:
\[ \text{HCF} \times \text{LCM} = 28 \times 30 \]
This rule saves us from factoring both numbers into primes separately.

Step 4: Solve for the LCM.
\[ 2 \times \text{LCM} = 28 \times 30 = 840 \]
\[ \text{LCM} = \frac{840}{2} = 420 \]
So the minimum number of books needed is 420.

Final Answer:
The minimum number of books to buy is 420, which corresponds to option (C). \[ \boxed{420} \]