Question

What is the least common multiple of 40 and 12?

Answer 1

Hint: To find the least common multiple of 40 and 12, we are going to first of all find the prime factorization of the two numbers. Then we are going to find the multiples which are common in both the prime factorizations of 40 and 12. After that we are going to multiply the not common factors from the two prime factorizations to the common multiples which we have found above.

Complete step by step solution:
In the above, we are going to find the least common multiple of the following numbers:
40 and 12
For that, we are going to find the prime factorization of both these numbers.
We are going to write the prime factorization of 40 and 12 one by one. First of all, we are going to find the prime factorization of 40.
$40=2\times 2\times 2\times 5$
Now, we are going to write the prime factorization of 12 as follows:
$12=2\times 2\times 3$
The factors which are common in both the prime factorizations of two numbers are as follows:
The factors which we have underlined in the above prime factorizations of 40 and 12 are as follows:
$40=\underline{2\times 2}\times 2\times 5$
$12=\underline{2\times 2}\times 3$
As you can see that the underlined factors in the above two prime factorizations are as follows:
$2\times 2=4$
The uncommon factors in the prime factorization of two numbers are as follows:
$2\times 5$ from 40 and 3 from 12 and we get,
$2\times 5\times 3=30$
Now, multiplying the common and the uncommon factors we get,
$4\times 30=120$

Hence, we got the least common multiple of 40 and 12 as 120.

Note: From the above solution, we can find the H.C.F (highest common factor). We know that H.C.F is the multiplication of the factors common in both the numbers 40 and 12 so in the above solution, the common factors which we have found are:
$2\times 2=4$
Hence, we have found the H.C.F of 40 and 12 as 4.