## vera_ewing one year ago How do you solve this?

1. vera_ewing

A particular integer N is divisible by two different prime numbers p and q. Which of the following must be true? I. N is not a prime number. II. N is divisible by pq. III. N is an odd integer.

2. anonymous

Do you know what a prime number is?

3. vera_ewing

A number that can only be divided by 1 and itself.

4. anonymous

yes so what do you think the answer is

5. vera_ewing

Well if the prime numbers p and q were 2 and 3, respectively, then that means I. would be incorrect right?

6. vera_ewing

Because then N would be 18, which is not a prime number.

7. vera_ewing

Ohh wait so I. would be correct! Right? @freckles

8. anonymous

yes

9. freckles

would there be any other options though ? for example if p and q are 2 and 3 respectively then you could say one possibly for N is 2(3)=6 doesn't 2(3) go into N=6? Also don't you mean I would be correct?

10. Here_to_Help15

First, recall that a prime number is only divisible by itself and 1, and that 1 is not a prime number. So, statement I must be true, since a number that can be divided by two prime numbers can’t itself be prime. Next, recall that every number can be written as a product of a particular bunch of prime numbers. Let’s say that N is divisible by 3 and 5. Then, N is equal to 3 ·5 ·p1 ·p2 · · ·, where p1, p2, etc. are some other primes. So, N is divisible by 3 · 5 = 15. Statement II must be true. Finally, remember that 2 is a prime number. So, N could be 6, since 6 = 2 · 3. Statement III isn’t always true.

11. freckles

Awesome explanation by @Here_to_Help15 :)

12. anonymous

agree

13. mathmath333

if N is divisible by two different prime numbers p and q then \large \color{black}{\begin{align} N\leq p\times q, \ \{p,q\} \in \mathbb{P} \hspace{1.5em}\\~\\ \end{align}} so N cannot be a prime number

14. anonymous

ok I get it

15. Here_to_Help15

Thanks :)

16. mathstudent55

@Here_to_Help15 Great explanation, just modify it this way (add the capitalized words): "Next, recall that every NON-PRIME number can be written as a product"