Question

The least common multiple of the numbers 20, 15 and 45 is

• A. 180
• B. 480
• C. 420
• D. 560

Hint:

To verify your LCM, check if it is divisible by the largest number. Since $180 \div 45 = 4$ and $180$ is also divisible by 20 and 15, it is a valid common multiple.

Answer 1

The Correct Option is A

Concept: The Least Common Multiple (LCM) of two or more numbers is the smallest positive integer that is divisible by all the given numbers. This can be calculated using the Prime Factorization method, where we take the highest power of every prime factor present in the numbers.

Step 1: Perform Prime Factorization for each number.
We break down each number into its constituent prime factors: \[ 20 = 2 \times 2 \times 5 = 2^2 \times 5^1 \] \[ 15 = 3 \times 5 = 3^1 \times 5^1 \] \[ 45 = 3 \times 3 \times 5 = 3^2 \times 5^1 \]

Step 2: Identify the highest powers of all prime factors.
The prime factors involved are 2, 3, and 5. We select the highest power for each:
• For the prime number 2, the highest power is $2^2$.
• For the prime number 3, the highest power is $3^2$.
• For the prime number 5, the highest power is $5^1$.

Step 3: Multiply these values to find the LCM.
\[ \text{LCM} = 2^2 \times 3^2 \times 5^1 \] \[ \text{LCM} = 4 \times 9 \times 5 \] \[ \text{LCM} = 36 \times 5 \] \[ \text{LCM} = 180 \] Final Answer: Option A