The LCM of 2x,3x and 5x (where x is a positive integer) is
The Correct Answer and Explanation is:
Answer:
The LCM of 2x, 3x, and 5x is 30x.
Explanation
To find the Least Common Multiple (LCM) of the algebraic expressions 2x2x, 3x3x, and 5x5x, we break the process into manageable steps. The LCM is the smallest expression that each of the given terms divides evenly into.
Step 1: Understand the Structure of the Terms
Each term contains a numerical coefficient and a variable part.
- 2x=2⋅x2x = 2 \cdot x
- 3x=3⋅x3x = 3 \cdot x
- 5x=5⋅x5x = 5 \cdot x
We treat coefficients and variables separately when finding the LCM.
Step 2: Find the LCM of the Coefficients
We need the LCM of the numbers 2, 3, and 5.
All of these are prime numbers. When finding the LCM of distinct primes, you multiply them: LCM(2,3,5)=2×3×5=30\text{LCM}(2, 3, 5) = 2 \times 3 \times 5 = 30
Step 3: Find the LCM of the Variable Part
Each term includes xx, and since all have exactly one factor of xx, we only need to take x1x^1 in the LCM.
If one term had x2x^2, for example, we would need to take the highest power of xx, but here all are x1x^1.
Step 4: Combine the Results
Now, combine the LCM of the coefficients (30) with the LCM of the variable part (xx): LCM(2x,3x,5x)=30x\text{LCM}(2x, 3x, 5x) = 30x
Final Answer:
30x\boxed{30x}
This is the smallest expression that each of the terms 2x2x, 3x3x, and 5x5x divides into without remainder. It includes all the factors needed to cover each of the original expressions.
