Question

Let $d\left(n\right)$ denotes the number of divisors of $n$ including $1$ and itself. Then, $d\left(225\right),d\left(1125\right)$ and $d\left(640\right)$ are

• A. In AP
• B. In HP
• C. In GP
• D. Consecutive integers

Answer 1

The correct option is C In GP

Here , $225=3^2\times 5^2$

$\Rightarrow d\left(225\right)=3\times 3=9$

$1125=3^2\times 5^3$

$\Rightarrow d\left(1125\right)=3\times 4=12$

and $640=2^7\times 5$

$\Rightarrow d\left(640\right)=8\times 2=16$

$\frac{12}{9}=\frac{16}{12}=\frac{4}{3}$

Hence, $9,12$ and $16$ are in GP.