SOLUTION: Choose all the descriptions for natural numbers n that have 3 divisors. 1- n= p^2 * q (for any two distinct primes p and q) 2- n= p * q (for any two distinct primes p and q

Question 1194009: Choose all the descriptions for natural numbers n that have 3 divisors.
1- n= p^2 * q (for any two distinct primes p and q)
2- n= p * q (for any two distinct primes p and q)
3- n= p * q * r (for any three distinct primes p,q and r)
4- n= p^2 (for any prime number)
So every natural number has at least 2 factors - 1 and itself. So numbers with 3 factors then have to be perfect squares of prime numbers.
I selected 4- n= p^2 (for any prime number)
So they only have 1 distinct prime factor, and the question says select ALL.
Am I missing any other description that applies?
Thanks

Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52776) (Show Source): You can put this solution on YOUR website!
.

If the question is

"Choose all the descriptions for natural numbers n that have / 3 divisors"

then your answer " option 4 " is correct.

As the question is worded in the post, it may have hidden EXTENDED meaning, and then the the answer " ALL " is applicable.

I understand that the " option 4 " is the first answer which comes to the mind,
but the answer " all options " is also possible, formally speaking.

So, this question/problem is consciously and ideally constructed in a way to catch (to cop)
a person at exam, with all the ensuing consequences . . .

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

Let's focus on n = p^2*q for now.

The unique primes are p and q
Though we have p show up twice in the form p^2

The divisors are:
1, p, q, p*q, p^2, p^2q
Clearly there are more than 3 divisors here so we move on.

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

With n = p*q, the divisors are:
1, p, q, p*q
We get close to 3 divisors, but we have one too many.

Now onto n = p*q*r
Divisors: 1, p, q, r, p*q, p*r, q*r, p*q*r
There are 8 divisors here, which we rule this answer choice out as well.
Thing to notice: There are 3 atomic pieces of p,q,r so there are 2^3 = 8 different divisors.
Why does this work? I'll leave it for you to think about. Hint: Think of the power set.

Lastly n = p^2
The divisors are: 1, p, p^2
This is exactly 3 divisors, so you have chosen the correct answer. This is the only answer that fits the 3 divisors pattern.

In summary, you are correct to think choice 4 is the only answer. The quick reasoning is that squaring any prime will have exactly 3 factors.

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Some concrete numeric examples:

For n = p^2*q we could pick p = 5 and q = 7
n = 5^2*7 = 175
Divisors: 1, 5, 7, 25, 35, 175
Number of divisors: 6

Now onto the form n = p*q
Let's go with p = 11 and q = 13
n = 11*13 = 143
Factors: 1, 11, 13, 143
Number of divisors: 4

Form: n = p*q*r
Let p = 2, q = 3, r = 7
n = 2*3*7 = 42
Divisors: 1, 2, 3, 6, 7, 14, 21, 42
Number of divisors: 8

Form: n = p^2
Let p = 17
n = p^2 = 17^2 = 289
Divisors: 1, 17, 289
Number of divisors: 3

A handy tool to use is WolframAlpha.
Type in something like "divisors of 289" and it will give the correct list in increasing order.

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