Question

N is a natural no. such that N=2^3×3^4×5^7×7^2×11 total no. of factors of N is
(1)-4×5×8×3×2
(2)-6×7×8
(3)-5×8×3×2
(4)-5×6×7×8×9
​

Answer 1

Answer:

$\boxed{\sf \:(1) \: \: \: 4 \times 5 \times 8 \times 3 \times 2 \: }$

Step-by-step explanation:

Given that, N is a natural number such that

$\sf \: N = {2}^{3} \times {3}^{4} \times {5}^{7} \times {7}^{2} \times 11$

We know,

If N is a composite number such that N = $\sf \: x^a y^b z^c$, where x, y, z are prime numbers and a, b, c are natural numbers, then number of factors is (a + 1)(b + 1)(c + 1).

So, using this result, we get

$\sf \: Number\:of\:factors$

$\sf \: = \: (3 + 1)(4 + 1)(7 + 1)(2 + 1)(1 + 1)$

$\sf \: = \: (4)(5)(8)(3)(2)$

$\sf \: = \: 4 \times 5 \times 8 \times 3 \times 2$

Hence,

$\implies\sf \: \boxed{\sf \: Number\:of\:factors = \: 4 \times 5 \times 8 \times 3 \times 2 \: }$

$\rule{190pt}{2pt}$

Additional Information

If N is a composite number such that N = $\sf \: x^a y^b z^c$, where x, y, z are prime numbers and a, b, c are natural numbers, then

$\sf \: Sum\:of\:factors = \bigg( \dfrac{ {x}^{a + 1} - 1}{x - 1} \bigg)\bigg( \dfrac{ {y}^{b + 1} - 1}{y - 1} \bigg)\bigg( \dfrac{ {z}^{c + 1} - 1}{z - 1} \bigg)$

$\sf \: Number\:of\:factors = (a + 1)(b + 1)(c + 1)$

$\sf \:Product\:of\:factors = {N}^{ \frac{Number\:of\:factors}{2} }$