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).

2 comments on “Number of Positive Divisors | Tomato objective 98”

Kaustav Roy says:

May 16, 2017 at 8:02 pm

Sir, can you plzz suggest me how to do TOMATO OBJECTIVE NO. 99 .i.e. Sum of all positive divisors of 1800

Reply
1. Anish Ray says:
  
  May 19, 2017 at 11:12 pm
  
  You can find the formula for the sum in any standard book namely An Excursion in Mathematics,Challenges and Thrills of Pre-College Mathematics.
  
  Or you may surf up the google
  Use the keyword-"Sum of divisors"
  
  Hope this helps
  
  Reply