Question

How many positive factors are there of the number 360?

Answer 1

Hint: In this question we need to find positive factors of a number that is $360$. We know that factors of a number $N$ refer to all the numbers which divide $N$ completely. There is a basic formula to find the number of positive factors of a number $N$.
$N = {a^p}{b^q}{c^r}$
Where $a, b$, and $c$ are prime factors of the number $N$.
$p, q$, and $r$ are non-negative exponents or power.
Number of factors of $N = (p + 1)(q + 1)(r + 1)$

Complete step by step solution:
In the very first step, we will find the factors of 360.
$ \Rightarrow 360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5$
$ \Rightarrow 360 = {2^3} \times {3^2} \times {5^1}$
Now, we will put the values in the above formula that is $N = (p + 1)(q + 1)(r + 1)$
Number of positive factors of $360 = (3 + 1)(2 + 1)(1 + 1) $
$= 4 × 3 × 2$
$ = 24$.

Hence, the number of positive factors of $360$ is $24$.

Note: Here, in the above question we were finding positive factors. In the first step, we have done the prime factorization. So, you must know how to factorize a number before solving this question. Prime factorization of a number is a method of breaking that number into its smaller parts. If we multiply all the prime factors together, we will get our original number. Once the factorization is done then we will express those factors in exponential form, For instance ${2^3} \times {3^2} \times {5^1}$. Then we will use the formula for finding the number of factors. We can also find the Product of all the factors and Sum of all the factors by using different formulae.