High School

Find the least common multiple of [tex]$10x$[/tex] and [tex]$45x^4$[/tex].

A. [tex]$90x^4$[/tex]
B. [tex]$450x^4$[/tex]
C. [tex]$90x^5$[/tex]
D. [tex]$5x$[/tex]

Please select the best answer from the choices provided:
A
B
C
D

Answer :

To find the least common multiple (LCM) of the expressions [tex]\(10x\)[/tex] and [tex]\(45x^4\)[/tex], we'll break it down into two parts: the numerical coefficients and the variables.

### Step 1: Find the LCM of the Numerical Coefficients
First, consider the numerical parts of the expressions, which are 10 and 45. To find the LCM of these numbers:

1. Prime Factorization:
- [tex]\(10 = 2 \times 5\)[/tex]
- [tex]\(45 = 3^2 \times 5\)[/tex]

2. Identify Common and Unique Factors:
- Here, the factors are: [tex]\(2, 3^2, 5\)[/tex].

3. Take the Highest Power of Each Prime Factor:
- [tex]\(2^1\)[/tex] from 10 (since 2 only appears in 10),
- [tex]\(3^2\)[/tex] from 45,
- [tex]\(5^1\)[/tex] from both (it's the same in both, so we take just one).

4. Calculate the LCM for the Coefficients:
[tex]\[
\text{LCM} = 2 \times 3^2 \times 5 = 2 \times 9 \times 5 = 90
\][/tex]

### Step 2: Combine with the Variable Part
Now consider the variable parts, [tex]\(x\)[/tex] and [tex]\(x^4\)[/tex]. We need to find the LCM in terms of the variable [tex]\(x\)[/tex]:

1. Identify the Highest Power of the Variable:
- In [tex]\(10x\)[/tex], the power of [tex]\(x\)[/tex] is [tex]\(1\)[/tex].
- In [tex]\(45x^4\)[/tex], the power of [tex]\(x\)[/tex] is [tex]\(4\)[/tex].
- The maximum power is [tex]\(x^4\)[/tex].

### Step 3: Combine the Results
Combine the LCM of the coefficients with the highest power of the variable:

[tex]\[
\text{LCM of } 10x \text{ and } 45x^4 = 90x^4
\][/tex]

Thus, the least common multiple of [tex]\(10x\)[/tex] and [tex]\(45x^4\)[/tex] is option (a) [tex]\(90x^4\)[/tex].