Question

Let $N=10800.$ Then the correct option(s) among the following is/are

• A. Number of even divisors is $48$
• B. Number of odd divisors is $12$
• C. Number of divisors that are multiple of $3$ is $45$
• D. Number of divisors that are multiple of $15$ is $24$

Answer 1

Answer: (A), (B), (C)

Number of divisors that are multiple of $3$ is $45$

Here $N=2^4\cdot 3^3\cdot 5^2$
Total number of divisors
$=(4+1)(3+1)(2+1)=60$

Now, the number of odd divisors
$=(3+1)(2+1)=12$

Then the number of even divisors
$=$ total divisors $-$ odd divisors
$=60-12=48$

For factors to be a multiple of $3$, we must select at least one $3$ which can be done in $3$ ways,
Therefore, the number of divisors that are multiple of $3$
$=(4+1)\times 3\times (2+1)=45$

We know that $15=3\times 5$
For factors to be a multiple of $15,$ we must select atleast one ${}^{\prime }{3}^{\prime }$ and atleast one ${}^{\prime }{5}^{\prime }$
Therefore, the number of divisors that are multiple of $15$
$=(4+1)\times 3\times 2=30$