If N = 2 5 × 3 x × 5 2 × 7 3 × 11 1 and the total number of facto

| If N = 2⁵ × 3^x × 5² × 7³ × 11¹ and the total number of factor of N is 576, then what is the value of x?

A. 1

B. 2

C. 3

D. 4

Please scroll down to see the correct answer and solution guide.

Right Answer is: C

SOLUTION

Formula used:

Let there be a composite number N and its prime factors be a, b, c, d, … etc. and p, q, r, s, … etc. be the indices (or powers) of the a, b, c, d, … etc respectively i.e., if N can be expressed as –

N = a^p. b^q.c^r.d^s…..

∴ The number of total factors of N is (p + 1) (q + 1) (r + 1)(s + 1) ….

Calculation:

N = 2⁵ × 3^x × 5² × 7³ × 11¹

⇒ (5 + 1) (x + 1) (2 +1) (3 + 1) (1 + 1) = 576

⇒ 6 × (x + 1) × 3 × 4 × 2 = 576

⇒ 144 × (x + 1) = 576

⇒ (x + 1) = 576/144

⇒ (x + 1) = 4

⇒ x = 4 – 1

⇒ x = 3

∴ The value of x is 3.