Question

Let $P=2^4\times 3^2\times 4^3\times 5^2\times 6^7,$ then the total number of divisors will be

• A. 180
• B. 360
• C. 540
• D. 720

Answer 1

The correct option is C 540

Given: $P=2^4\times 3^2\times 4^3\times 5^2\times 6^7$

If any natural number N can be written as the product of prime numbers by prime factorisation method

i.e. $N=Y^B\times Y^D\times Z^D,$ where X, Y and Z are prime numbers and a, b and c are their respective powers, then, the total number of factors of N is given by (a + 1) (b + 1) (c + 1).

Now, here $P=2^4\times 3^2\times 4^3\times 5^2\times 6^7$

$=2^4\times 3^2\times {(2\times 2)}^3\times 5^2\times {(2\times 3)}^7$

$=2^4\times 3^2\times 2^3\times 2^3\times 5^2\times 2^7\times 3^7$

$=2^{17}\times 3^9\times 5^2$

Here, 2, 3 and 5 are primes and 17, 9 and 2 are their respective powers. Therefore, the total number of divisors are (17 + 1) × (9 + 1) × (2 + 1) = 18 × 10 × 3 = 540

Hence, the correct answer is option (3).