Question

How do you solve this?

Answer 1

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.

Answer 2

Do you know what a prime number is?

Answer 3

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

Answer 4

yes so what do you think the answer is

Answer 5

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

Answer 6

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

Answer 7

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

Answer 8

yes

Answer 9

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?

Answer 10

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.

Answer 11

Awesome explanation by @Here_to_Help15
:)

Answer 12

agree

Answer 13

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

Answer 14

ok I get it

Answer 15

Thanks :)

Answer 16

@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"