ISI2014-DCG-10

Question

The number of divisors of $6000$, where $1$ and $6000$ are also considered as divisors of $6000$ is

• A. $40$
• B. $50$
• C. $60$
• D. $30$ 

Answer 1

6000 can be written as 2^4 * 5^3 *3.

therefore any divisor of 6000 can be formed by taking any combination of the above factors.

so for '2' there are 5 choices whether to include it or not in the divisor of 6000.

similarly for '5' there are 4 choices and for '3' there are 2 choices.

therefore total no of choices = 5*4*2 = 40.

there (A) is the correct answer.

Answer 2

N= $6000$, we can write it in form of multiple of co-primes

N= $2^4* 3^1* 5^3$

No of divisors = $(4+1)(1+1)(3+1)= 40$

Option A) is correct

Comments

"we can write it in form of multiple of co-primes"

Actually they are primes. Every positive integer ($>1$) can be written as the product of primes having positive powers which is known as Fundamental Theorem of Arithmetic.

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2^4 *3* 5^3 

 

No of divisors = (5+1)(1+1)(3+1)=40

Option A is correct

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40 will be the correct answer here.

Answer 3

https://www.math.upenn.edu/~deturck/m170/wk2/numdivisors.html

You can find the factors as 6000 = $2^{4}*3*5^{3}$ then as per trick you add 1 to each exponent and multiply them, ie,

{4+1}*{1+1}*{3+1}  = 40