Question

How do you find the least common multiple of $3,4w + 2$ and $4{w^2} - 1$?

Answer 1

Hint: LCM: Least Common Multiple : Least common multiple can be found by multiplying the highest exponent prime factors of $3,4w + 2$ and $4{w^2} - 1$ . For this we have to calculate the prime factor of $3,4w + 2$ and $4{w^2} - 1$ . Least common multiple is also known as lowest common denominator and this can be calculated in two ways ; with the help of greatest common factor (GCF), or multiplying the prime factors with the highest exponent factor.

Complete step-by-step solution:
Given $3,4w + 2$ and $4{w^2} - 1$
Now we know we have to do factorization of $3,4w + 2$ and $4{w^2} - 1$ .
Prime factorization: It’s the process where the original given number is expressed as the product of
prime numbers.
Factorization of $3$ :
$\Rightarrow 3 = 3$
 Factorization of $4w + 2$ :
$\Rightarrow 4w + 2 = 2(2w + 1)$
Factorization of $4{w^2} - 1$ :
$\Rightarrow 4{w^2} - 1 = (2w - 1)(2w + 1)$
Least common multiple can be found by multiplying the highest exponent factors of $3,4w + 2$ and $\Rightarrow 4{w^2} - 1$ .
Therefore, we can write ,
LCM of $3,4w + 2$ and $4{w^2} - 1$ is given as ,
$ \Rightarrow 2 \times 3 \times (2w - 1) \times (2w + 1)$

Therefore, we can write that the LCM of $3,4w + 2$ and $4{w^2} - 1$ is $2 \times 3 \times (2w - 1) \times (2w + 1)$.

Note: Questions similar in nature as that of above can be approached in a similar manner and we can solve it easily. The above question can also be done by simply listing all the possible factors of both the given numbers and then taking the greatest common factor by just comparing the list of factors to find the common factors and choosing the greatest factor among it then applying the formula that is LCM is equals to the product of the HCF and the product of the numbers.