Question

The exponent of 2 in the prime factorization of 144 is
$\left(a\right)4$
$\left(b\right)5$
$\left(c\right)6$
$\left(d\right)3$

Answer 1

Hint: We are required to find the exponent of a prime that is 2 in the prime factorization of 144. For that we first need to find the prime factorization of the number i.e. we need to repeatedly divide the number by the smallest prime possible and we need to that till it becomes 1. So, in short we need to find the prime numbers which when multiplied will give the number whose prime factorization is needed to be found out.

Complete step by step answer:
So, we first find the prime factorization of the number, for which we will continuously divide the number by the smallest prime possible. For example, if the number is 6 then we first find the smallest prime that divides 6, which is 2 and then we divide it further by 3 because that is the smallest prime which divides 6 after 2. And now only 1 is left which means the prime factorization of 6 is as follows:
$6=2\times 3\times 1$
Now, we have 144. The smallest prime number that divides 144 is 2, so we have:
$144=2\times 72$
Now, we further factorize 72. The smallest prime number that divides 72 is 2 again, so till now we have reached:
$144=2\times 2\times 36$
Again, 2 divide 36, so we have:
$144=2\times 2\times 2\times 18$
$\Rightarrow 144=2\times 2\times 2\times 2\times 9$
Now, 3 is the smallest prime that divides 9, so:
$\Rightarrow 144=2\times 2\times 2\times 2\times 3\times 3$
So, the prime factorization of 144 is found since no prime can be decomposed further. So, we can write:
$\Rightarrow 144={{2}^{4}}\times {{3}^{2}}$

And it can easily be seen that the exponent of 2 in the prime factorization of 144 is 4. Hence, option $\left(a\right)4$ is correct.

Note: You need to be very careful while giving the factors, because only prime factors are allowed while finding the prime factorization. For example, if you write $256=2\times 2\times 2\times 32$ then that would be wrong because 32 is not a prime number. So, you need to be aware while listing out the prime factors.