Question

The LCM of 960 and 240 is :

• A. 960
• B. 240
• C. 60
• D. 15

Hint:

Whenever you are asked to find the LCM of two numbers, always check if the larger number is divisible by the smaller one first.
If it is, the larger number is the LCM, and the smaller number is the HCF.
This simple observation can save you from writing out prime factorizations during the exam!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks us to find the Least Common Multiple (LCM) of two positive integers, 960 and 240.

Step 2: Key Formula or Approach:
1. The Least Common Multiple (LCM) of two or more numbers is the smallest positive integer that is perfectly divisible by all the given numbers.
2. If one number is a direct integer multiple of another, then the larger number is automatically the LCM of the two numbers.
3. Alternatively, we can find the LCM using the prime factorization method:
\[ \text{LCM} = \text{product of the highest powers of all prime factors involved.} \]

Step 3: Detailed Explanation:
1. Let us analyze the two given numbers, \(a = 960\) and \(b = 240\).
2. We can check if 960 is divisible by 240:
\[ \frac{960}{240} = 4 \]
Since 960 is a perfect multiple of 240 (i.e., \(960 = 240 \times 4\)), the smallest number that is a multiple of both 960 and 240 is 960 itself.
3. We can also verify this using Prime Factorization:
- Find the prime factors of 240:
\[ 240 = 24 \times 10 = (2^3 \times 3) \times (2 \times 5) = 2^4 \times 3^1 \times 5^1 \]
- Find the prime factors of 960:
\[ 960 = 96 \times 10 = (2^5 \times 3) \times (2 \times 5) = 2^6 \times 3^1 \times 5^1 \]
4. To find the LCM, take the highest power of each prime factor present in either factorization:
- For prime factor 2, the highest power is \(2^6\).
- For prime factor 3, the highest power is \(3^1\).
- For prime factor 5, the highest power is \(5^1\).
Multiply these values together:
\[ \text{LCM}(960, 240) = 2^6 \times 3^1 \times 5^1 = 64 \times 3 \times 5 = 64 \times 15 = 960 \]
Both methods yield the same result of 960.

Step 4: Final Answer:
The LCM of 960 and 240 is 960, which corresponds to option (A).