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Question

Let P=24×32×43×52×67, then the total number of divisors will be

A
180
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B
360
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C
540
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D
720
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Solution

The correct option is C 540
Given: P=24×32×43×52×67

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

i.e. N=YB×YD×ZD, 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=24×32×43×52×67

=24×32×(2×2)3×52×(2×3)7

=24×32×23×23×52×27×37

=217×39×52

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).

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