SOLUTION: Let $m$ be a positive integer. If $m$ has exactly $18$ positive divisors, then how many positive divisors does $m^2$ have?

Question 1207622: Let $m$ be a positive integer. If $m$ has exactly $18$ positive divisors, then how many positive divisors does $m^2$ have?
Found 2 solutions by greenestamps, math_tutor2020:
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

There is not a single answer with just the given information.

m has 18 positive divisors. There are four possibilities:

(1) 18 = 18*1; --> --> m^2 has 35 positive divisors

(2) 18 = 9*2; --> --> m^2 has 17*3 = 51 positive divisors

(3) 18 = 6*3; --> --> m^2 has 11*5 = 55 positive divisors

(4) 18 = 3*3*2; --> --> m^2 has 5*5*3 = 75 positive divisors

Answer by math_tutor2020(3816) (Show Source): You can put this solution on YOUR website!

The prime p has exactly two divisors: 1 and itself.
Example: the prime 7 has factors 1 and 7 only.

p^k has k+1 divisors and they are:
p^0, p^1, p^2, ..., p^(k-1), p^k
Note that p^0 = 1 and p^1 = p.
Example: p = 3 and k = 4 yields p^k = 3^4 = 81 having k+1 = 4+1 = 5 divisors.
Those five divisors are 1, 3, 9, 27, 81.

Let p and q represent two different primes.
Let k and m be positive integers.
We can construct the following composite number
n = p^k*q^m

How many divisors does n have?
Imagine we can form a table that has k+1 rows and m+1 columns.
Along the left hand side we will have the labels p^0, p^1, p^2, ..., p^(k-1), p^k.
Along the top header we will have the labels q^0, q^1, q^2, ..., q^(m-1), q^m.

This table generates (k+1)*(m+1) different divisors.
Each entry into the table is the result of multiplying the headers. Eg: multiply p^2 with q^3 to get p^2q^3 as one divisor of n.

This logic can be extended to as many unique prime factors as you want.
Example: n = p^k*q^m*r^s will have (k+1)*(m+1)*(s+1) divisors.
We add 1 to each exponent, then multiply those results.

More info found here
https://mathworld.wolfram.com/DivisorFunction.html
Refer specifically to equation (3) on that page.

--------------------------------------------------------------------------

So this is why tutor greenestamps had written various ways to factor 18.
His goal is to write an equation like
18 = (k+1)(m+1)
or
18 = (k+1)(m+1)(s+1)

Let me know if you have any questions.

RELATED QUESTIONS
Let $m$ and $n$ be positive integers. If $m$ has exactly $10$ positive divisors, $n$ has (answered by greenestamps)
Let $M$ be the least common multiple of $1,$ $2,$ $\dots,$ $12$, $13$, $14$, $15$, $16$. (answered by greenestamps)
A positive integer is called terrific if it has exactly $3$ positive divisors. What is... (answered by greenestamps)
A positive integer is called terrific if it has exactly $3$ positive divisors. What is... (answered by josgarithmetic)
A positive integer is called terrific if it has exactly $3$ positive divisors. What is (answered by greenestamps)
What is the smallest positive integer that has exactly 7 positive divisors? (answered by jim_thompson5910)
How many positive integers have exactly 3 proper divisors, each of which is less than 50? (answered by Edwin McCravy,CarlosOrtiz)
what is the smallest positive integer with exactly five divisors... (answered by richwmiller)
The number $100$ has four perfect square divisors, namely $1,$ $4,$ $25,$ and $100.$... (answered by greenestamps)