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Number of Positive Divisors | Tomato objective 98

This is an objective problem from TOMATO based on finding the Number of Positive Divisors.

Problem:

The number of positive integers which divide $240$ is-
(A) $18$; (B) $20$; (C) $30$; (D) $24$;

Discussion:

We use the formula for computing number of divisors of a number:

Step 1: Prime factorise the given number

$240 = 2^4 \times 3^1 \times 5^1 $

Step 2: Use the formula for number of divisors: $(4+1) \times (1+1) \times (1+1) = 20 $

Answer: (B) 20;

Note:

Why this formula works? Basically, we are adding 1 to each exponent of each prime factor and then multiplying them. Refer to a discussion on Number Theoretic Functions (in any standard number theory book like David Burton's Elementary Number Theory).

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