The number of divisors of apbqcrds where a- b- c- d are primes p- q-r- s - N- excluding 1 and the number itself- is-

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The number of...

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The number of divisors of $a^{p} b^{q} c^{r} d^{s}$ where $a, b, c, d$ are primes & $p, q, r, s$ $ϵ$ N, excluding $1$ and the number itself, is:

p q r s

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(p + 1) (q + 1) (r + 1) (s + 1) - 4

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p q r s - 2

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(p + 1) (q + 1) (r + 1) (s + 1) - 2

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If $p > q > 0 a n d p r < - 1 < q r$ , then find the value of

{tan}^{- 1} \frac{p - q}{1 + p q} + {tan}^{- 1} \frac{q - r}{1 + q r} + {tan}^{- 1} \frac{r - p}{1 + r p} .

\begin{matrix} List - I & List - II (P) & \frac{\prod_{k = 1}^{100} (1 - t a n^{2} \frac{2^{k} π}{2^{100} + 1})}{2^{100}} is & 1. & 1 (Q) & \prod_{k = 1}^{100} (1 + 2 c o s \frac{2 π 3^{k}}{3^{100} + 1}) is & 2. & - 1 (R) & Let n be a positive integer and & 3. 2 Let x be a real number different from 2^{k + 1}, k = 1, \dots \dots n, then [(1 + 2 c o s \frac{x}{2^{100}}) \prod_{k = 1}^{100} (1 - 2 c o s \frac{x}{2^{k}})] - 2 c o s x is equal to (S) & The value of \sum_{n = 1}^{\infty} \frac{1}{2^{n}} t a n (\frac{2}{2^{n}}) + c o t 2 i s & 4. & 3 5. & \frac{1}{2} \end{matrix}

(p^{2} + p + 1) (q^{2} + q + 1) (r^{2} + r + 1) (s^{2} + s + 1) \geq 81 p q r s

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