If prime numbers p and t are the only prime factors

[SummerCourse]

Any input on this problem?

[StaceyKoprince]

Sorry this one got lost in our reorganizing shuffle. Also, please try to reserve image files for problems with complicated diagrams or the like. It takes longer to download and view image files and we're trying to get through as many questions as we can - we need your help to make it efficient!

If the prime numbers p and t are the only prime factors of the integer m, is m a multiple of t*p^2?
(1) m has more than 9 positive factors
(2) m is a multiple of p^3

Draw a prime box and put p and t inside. According to the problem, there could be multiple instances of p and t in there, but that's it. We want to know whether there are at least two p's and one t in there.

Start with statement 2. If m is a multiple of p^3, that means there are 3 p's in m's prime box. There's already a t in there, according to the original question. So there are at least 2 p's and one t. Answer to question is yes, so statement is sufficient. Eliminate A, C, E.

Statement 1. Notice that this just says "positive factors" NOT prime factors. The complete set of factors is made by multiplying the prime factors in different combinations. For example, 12 has the prime factors 2, 2, and 3. We can find all of the general factors of 12 by taking 2, 3, 2*2, 2*3, 2*2*3, and of course 1.

So m has more than 9 positive factors. Well, I know m has p and t - there are 2 factors. And I know m has 1 and itself - there are 2 more factors, for a total of four. I need five more, so I have to add to my prime box to be able to create five more general factors. The only things I can put in my prime box are p and t. I can put all p's, all t's, or some combination of p's and t's. If I put in at least one p, then I'd have at least 2 p's and one t, which would answer the question "yes." BUT, if I put in all t's, then I'd only have one p, which would answer the question "no" - so the statement is insufficient.

[Jazmet]

Hi Stacey,

Could you please help me with the statement 2 by trying numbers.

p = 3
t = 2

Q - Is 'm' a multiple of 18?

statement 2; 'm' = 27 , 54 -

In this case 27 is not a multiple of 18 and 54 is.

Please help.

[RonPurewal]

Hi Stacey,

Could you please help me with the statement 2 by trying numbers.

p = 3
t = 2

Q - Is 'm' a multiple of 18?

statement 2; 'm' = 27 , 54 -

In this case 27 is not a multiple of 18 and 54 is.

Please help.

27 is not a valid case.
The conditions state that both p and t are factors of m. With your numbers, that means 2 and 3 are both factors of m.
Along with statement 2, that means 2 and 27 are both factors of m.

So, 54, 108, etc. You'll see pretty quickly that you're looking at multiples of 54.

[Jazmet]

Ohhh! got it. Thank you Ron for your swift reply.

It was a silly mistake!

[RonPurewal]

Ohhh! got it. Thank you Ron for your swift reply.

It was a silly mistake!

It may be a "silly" mistake, but it's still a mistake. And it deserves just as much attention as other types of mistakes -- if not more attention (since you'll have the opportunity to make this type of mistake quite often, if you don't remedy it).

Note it -- and note which situations deserve extra attention in future problems.

[Jazmet]

Sure! Thank you.

[RonPurewal]

Sure! Thank you.

you're welcome