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Q.
The number of divisors of $6912, 52488, 32000$ are in
Sequences and Series
Solution:
Fact: If $n$ is $a +ve$ number and $n =P^{k_{1}}_{1} \cdot P_{2}^{k_{2}} ...P^{k_{r}}_{r}$
(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 $1$.