Question

The total number of divisors of 480, that are of the form 4n + 2, $n\ge 0$, is equal to
(a) 2
(b) 3
(c) 4
(d) None of these

Answer 1

In order to solve this question, we should know that the number of the divisor of any number $x={{a}^{m}}{{b}^{n}}{{c}^{p}}....$ where a, b, c are prime numbers and is given by (m + 1) (n + 1) (p + 1)….. By using this property we can find the solution of this question.

Complete step-by-step answer:
In this question, we have been asked to find the total number of divisors of 480 which are of the form 4n + 2, $n\ge 0$. To solve this question, we should know that the total number of divisors of any number x of the form ${{a}^{m}}{{b}^{n}}{{c}^{p}}.....$ where a, b, c … are prime numbers and is given by (m + 1) (n + 1) (p + 1)….. we know that 480 can be expressed as
$480={{2}^{5}}.3.5$
So, according to the formula, the total number of divisors of 480 are (5 + 1) (1 + 1) (1 + 1) = $6\times 2\times 2=24$.
Now, we have been asked to find the number of divisors which are of the form 4n + 2 = 2 (2n + 1), which means odd divisors cannot be a part of the solution. So, the total number of odd divisors that are possible are (1 + 1) (1 + 1) = $2\times 2=4$, according to the property.
Now, we can say the total number of even divisors are = all divisors – odd divisor
= 24 – 4
= 20
Now, we have been given that the divisor should be of 4n + 2, which means they should not be a multiple of 4 but multiple of 2. For that, we will subtract the multiple of 4 which are divisor of 480 from the even divisors.
And, we know that, $480={{2}^{2}}\left( {{2}^{8}}.3.5 \right)$. So, the number of divisors that are multiples of 4 are (3 + 1) (1 + 1) (1 + 1) = $4\times 2\times 2$ = 16. Hence, we can say that there are 16 divisors of 480 which are multiple of 4.
So, the total number of divisors which are even but not divisible by 2 can be given by 20 – 16 = 4.
Hence, we can say that there are 4 divisors of 480 that are of 4n + 2 form, $n\ge 0$.
Hence, the option (c) is the right answer.

Note: We can also solve this question by writing 4n + 2 = 2(2n + 1) where 2n + 1 is always an odd number. So, when all odd divisors will be multiplied by 2, we will get the divisors that we require. Hence, we can say a number of divisors of 4n + 2 form is the same as the number of odd divisors for 480.