Question

Find the smallest positive integer such that it has exactly $100$ different positive divisors including 1 and the number itself.

• A. $45360$
• B. $45370$
• C. $45380$
• D. $45390$

Answer 1

The correct option is A $45360$

We know That total No. of factors

$=productof(primen{o}^{\prime }spower+1)$

If $N$ is the number of different divisors:

$N=(p_1+1)\cdot (p_2+1)\cdot \cdot \cdot (p_n+1)$

$100=2^2\times 5^2$

$=2\times 2\times 5\times 5=(1+1)(1+1)(4+1)(4+1)$

Then the integer $n=$ $a_1^{p_1}\cdot a_2^{p_2}\cdot \cdot \cdot \cdot a_n^{p_n}$

For the smallest value: $p_1=4,p_2=4,p_3=1,p_4=1$

Then,

$n=a_1^4\times a_2^4\times a_3^1\times a_4^1$

$=2^4\cdot 3^4\cdot 5^1\cdot 7^1$

$=16\cdot 81\cdot 5\cdot 7$

$=45360$

Hence, the smallest number is $45360.$