What is the smallest integer with 100 non trivial factors?

Question

Excluding 1 and itself as factors, what is the smallest integer with 100 different factor values?

Example: 20 has 4 non trivial factors (2*10 and 4*5)

Answer 1

Assuming you mean "exactly $100$ non-trivial divisors".

If the natural number $n>1$ has prime decomposition $n=\prod p_i^{a_i}$, the total number of factors is $$d(n)=\prod (a_i+1)$$

As you wish to exclude $1$ and $n$, you are interested in $F(n)=d(n)-2$. We see that you are requiring $F(n)=100$ or, equivalently, $d(n)=102$

Now, $102=2\times 3\times 17$. We therefore seek $\{a_i\}$ with $\prod (a_i+1)=2\times 3\times 17$. We remark that our $n$ can not have more than three prime divisors.

Case I: One prime. Then the best we can do is $n=2^{101}$ which seems unlikely to be minimal.

Case II. Two primes Then the best we can do is $2^{16}3^5$, certainly a lot better than $2^{101}$

Case III. Three primes. Now the best we can do is $2^{16}3^25$, which is only $\boxed {2949120}$.

Answer 2

This can be found with OEIS entry A061799, for which $a_n$ is the smallest number with at least $n$ divisors. Your definition excludes 1 and the number itself, so we're looking for $a_{102}$ which is 50400.

Alternatively, use OEIS entry A005179, for which $a_n$ is the smallest number with exactly $n$ divisors. In this case $a_{102}$ is 2949120.

Answer 3

If it has to be 100 different prime factors, the 10 year olds in my class looked at the pattern of answers starting at 1 prime factor, 2,3,4, 5,6 etc and the size of the numbers (powers of $10$). We decided that as a minimum the answer must have at least $98$ zeros.