Question

The number of divisors of $6912,52488,32000$ are in

• A. $A.P.\&H.P$.
• B. $GP.\&H.P$.
• C. $A.P.\&GP$.
• D. None of these

Answer 1

Answer: (C)

Fact: If $n$ is $a+ve$ number and $n=P_1^{k_1}\cdot P_2^{k_2}...P_r^{k_r}$
(where $p_1,p_2,p_3,...p_r$ are prime numbers) then number of divisors of n are given by
$\left(k_1+1\right)\left(k_2+1\right)...\left(k_r+1\right)$
Prime factorisation of $6912=2^8\cdot 3^3$
$\therefore$ Number of divisors $=9\times 4=36$
Prime factorisation of $52,488=3^8\times 2^3$
$\therefore$ No. of divisors $=9\times 4=36$
Prime factorisation of $32,000=5^3\times 2^8$
$\therefore$ No. of divisors $=9\times 4=36$
Now, each number having same number of divisors i.e., $36,36,36$
Each and every term is constant & constant sequence is always in $A.P.\&G.P$. both, as common difference is $0$ and common ratio is $1$.