Question

Let $N=10800$

$\text{I}.$ Total number of divisors of $N$ is $60$
$\text{II}.$ The number of even divisors of $N$ is $24$
$\text{III}.$ The number of divisors of $N$ which are multiples of $15$ is $30$

Then which of the following statements are true?

• A. only $\text{I}$
• B. only $\text{I and}\text{III}$
• C. only $\text{I and}\text{II}$
• D. All of the above

Answer 1

The correct option is B only $\text{I and}\text{III}$

$N=10800=2^4\times 3^3\times 5^2$
Any divisor is of the form $2^a{3}^b{5}^c$ where $a\in \{0,1,2,3,4\}$
$b\in \{0,1,2,3\}$ and
$c\in \{0,1,2\}$

$\therefore$ Total number of divisors $=(4+1)(3+1)(2+1)=60$

For divisors to be even, we must have at least one $2$ out of four $2^{\prime }\text{s},$ any number of $3^{\prime }\text{s}$ and $5^{\prime }\text{s}$
So, required number $=4\times (3+1)(4+1)=48$

A divisor will be multiple of $15$ if it contains at least one $3$ out of three $3^{\prime }\text{s}$ and at least one $5$ out of two $5^{\prime }\text{s}$
So, required number $=(4+1)\times 3\times 2=30$