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: Understanding the Question:
There are two sections, A and B, containing 28 and 30 students respectively. We need to find the minimum number of books required for the library such that they can be divided equally among either Section A (28 students) or Section B (30 students) without any remainder.

Step 2: Key Formula or Approach:
The minimum number of books that is exactly divisible by both 28 and 30 is their Lowest Common Multiple (L.C.M.). We can find this using the prime factorization method:
\[ \text{L.C.M.} = \text{product of the highest powers of all prime factors involved.} \]

Step 3: Detailed Explanation:
1. Find the prime factorization of both numbers:
- For 28:
\[ 28 = 4 \times 7 = 2^2 \times 7^1 \]
- For 30:
\[ 30 = 2 \times 3 \times 5 = 2^1 \times 3^1 \times 5^1 \]
2. Identify the highest power of each prime factor present in either factorization:
- For prime factor 2: the highest power is \(2^2\).
- For prime factor 3: the highest power is \(3^1\).
- For prime factor 5: the highest power is \(5^1\).
- For prime factor 7: the highest power is \(7^1\).
3. Multiply these highest powers together to find the L.C.M.:
\[ \text{L.C.M.}(28, 30) = 2^2 \times 3^1 \times 5^1 \times 7^1 \]
\[ \text{L.C.M.}(28, 30) = 4 \times 3 \times 5 \times 7 \]
\[ \text{L.C.M.}(28, 30) = 12 \times 35 \]
\[ \text{L.C.M.}(28, 30) = 420 \]
This means the minimum number of books that can be distributed equally among students of either section is 420.

Step 4: Final Answer:
The minimum number of books to acquire is 420, which corresponds to option (C).