Question

The number of factors of 1080 is?
(a) 32
(b) 28
(c) 24
(d) 36

Answer 1

Hint: First understand the meaning of the term ‘factor’ of a number. Now, to find the number of factors of 1080 write it as the product of its primes. Group the similar prime factors and write them in the exponential form as ${{a}^{m}}\times {{b}^{n}}\times {{c}^{p}}...$ and so on. Here a, b, c…. are the different prime factors and m, n, o… are their exponents. Use the formula for the number of prime factors given as $\left( m+1 \right)\left( n+1 \right)\left( p+1 \right)....$ to get the answer.

Complete step-by-step answer:
Here we have been provided with the number 1080 and we are asked to find the number of factors it has. First we need to understand the term ‘factor’.
Now the term ‘factor’ (x) of a number (y) means x divides y completely without leaving any remainder. For example: 18 can be divided by 2, 3, 6 and 9 without leaving any remainder, so they are the factors of 18. Also, 1 is a factor of every number and a number is a factor of itself so we can say 1 and 18 are also the factors of 18.
Let us come to the question. To determine the number of factors of 1080 first we need to write it as the product of its prime factors, so we get,
\[\Rightarrow 1080=2\times 2\times 2\times 3\times 3\times 3\times 5\]
In the exponential form we can write the above prime factorization as: -
\[\Rightarrow 1080={{2}^{3}}\times {{3}^{3}}\times {{5}^{1}}\]
Now, a number of the form ${{a}^{m}}\times {{b}^{n}}\times {{c}^{p}}...$ has $\left( m+1 \right)\left( n+1 \right)\left( p+1 \right)....$ factors, so we can say that the number 1080 has $\left( 3+1 \right)\left( 3+1 \right)\left( 1+1 \right)$ factors.
$\Rightarrow $ Number of factors of 1080 = $4\times 4\times 2$
$\Rightarrow $ Number of factors of 1080 = 32

So, the correct answer is “Option A”.

Note: Note that here we do not have to find the prime factors only but we have to find all the factors whether prime or composite. To find the composite factors we need to group different prime factors. It is difficult to form groups of different prime factors by considering their product because factors are repeated several times. That is why we need to remember the formula used for the calculation of the total number of factors.