searching smallest number that has $40$ distinct positive divisors What is the smallest natural number such that it has $ 40 $ distinct positive (integer) divisors (inclusive of $ 1 $ and itself?
At first I was stunned of seeing the problem.It's not possible to find all the divisors of all the numbers. I think it's  a huge calculation. How can I solve this type of problem easily?  help me.How can I proceed? 
 A: Letting $$N={p_1}^{q_1}\cdot {p_2}^{q_2}\cdots \cdot {p_k}^{q_k}\ \ (p_i\ \text{are primes}, q_i\ge 1\in\mathbb N,k\in\mathbb N)$$ be your number, the following has to be satisfied (see here for details):
$$(q_1+1)(q_2+1)\cdots(q_k+1)=40=2^3\cdot 5.$$
(Here, LHS represents the number of the positive divisors of $N$.)
So, separate it into cases as the followings :
(1) Since $40=40$, $N=p^{39}$. Hence, we have $2^{39}.$ (Note this is the smallest number in this case)
(2) Since $40=2\times 20$, $N={p_1}^{1}\cdot {p_2}^{19}$. Hence, we have $3^1\cdot 2^{19}.$
(3) Since $40=4\times 10$, $N={p_1}^{3}\cdot {p_2}^{9}$. Hence we have $3^3\cdot 2^9$.
Can you take it from here? Note that there are still several cases.
A: Hint: Consider first how to determine the number of distinct divisors of a number from its prime factorisation.
So any prime has two factors, namely $1$ and itself.
If you can do this, you will be able to find numbers with $40$ factors quite easily - but there will be a little work to do to find the smallest.
A: Stated a slightly different way, consider the function d(n), which is the gives the number of  divisors for integer n. d(n) is multiplicative, in that if the prime decomposition of n is:
$$
n = {p_1}^{q_1} * {p_1}^{q_2} \cdots \cdot {p_k}^{q_k}\ \ p_i\ \text{are primes} 
$$
The number of divisors in n is the product of the number of divisors in each factor of n.
$$
d(n) = d({p_1}^{q_1}) * d({p_1}^{q_2}) \cdots \cdot d({p_k}^{q_k})\ \ p_i\ \text{are primes} 
$$ 
Remembering that the number of divisors in ${p_1}^{q_i}$ is $q_i + 1$ (since it includes 1 and itself) this should be a good start, as mathlove stated. You're looking for the product to be 40, so you have several combinations to try out (such as 4*10, 8*5, 5*4*2, etc).
Good luck! 
