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If the prime numbers p and t are the only prime factors of the integer m, is m a multiple of (p^2)(t) ?

1) m has more than 9 positive factors

2) m is a multiple of p^3

Solution:

The problem states that p and t are the only prime factors of the integer m. which means m can be written as

m = (p^^i) * (t^^j) where i and j are any two integers, i,j>0.

Note that m may not necessirily be equal to p*t. It is one of the many possibilities and m = pt is a special case when i=j=1.

The problem boils down to finding whether i >= 2.

1. Also, note that the # of factors of m = (i+1) * (j+1) = 9. This does not tell you whether i>= 2, because i can be equal to 1 and j >= 4 to satisfy the above condition. Hence, this statement alone is not sufficient.

2. This is better, it says that p^^3 is a multiple of m, hence i must be greater than 2, and hence you can say that (p^2)(t) is a multiple of m as well.

thanks srinivasssrk, however, i am having difficulty following your explanation due to my inferior math skills. Is there a simplier way to explain this?

Given that m has only two prime factors. I think where people have trouble is understanding the difference between a prime factor and a factor. A factor is any integer that divides a number evenly (with zero remainder). Where as a prime factor is a factor that is a prime number as well.

examples:

a. even though both 2 and 4 are factors of 8, only 2 is the prime factor.
b. even though 2, 3, 4 and 6 are all factors of 12, only 2 and 3 are the prime factors.

Note also that even though 8 (=2*2*2) has only one prime factor i.e 2, it has other factors (4 which is a multiple of 2 and is also 2^2).

Similarly, 12 is expressed as 2*2*3, note that only 2 and 3 are the prime factors, but it also has other factors 4 and 6.

Now, coming to the problem, m has two prime factors p and t.

Again, using some numbers as examples, let us pick 2 and 3 as p and t.

Now, can you say m = 2 *3 = 6. No. because m can be any integer which is multiples of just 2 and 3. For example, m can be

Now, you are asked to find out whether 2^2*3 (p^2 * t) is a multiple of m.

1. Statement 1 says that m has more than 9 factors.

m will have more than 9 factors depending on howmany 2's and 3's you pick from above. You can necessirily say there will be two 2's (p^2 right, you need atleast two p's as factors two say p^2 is a factor).

2. Where as statement 2 is straight forward, it says p^^3 is a factor and hence p^^2 should be a factor as well.

(in the above logic substutite 2's and 3's from the above example).

Let me know if I confused you more. I will try to present it in a different way.

If the prime numbers p and t are the only prime factors of the integer m, is m a multiple of (p^2)(t) ?

1) m has more than 9 positive factors

2) m is a multiple of p^3

Solution:

The problem states that p and t are the only prime factors of the integer m. which means m can be written as

m = (p^^i) * (t^^j) where i and j are any two integers, i,j>0.

Note that m may not necessirily be equal to p*t. It is one of the many possibilities and m = pt is a special case when i=j=1.

The problem boils down to finding whether i >= 2. 1. Also, note that the # of factors of m = (i+1) * (j+1) = 9. This does not tell you whether i>= 2, because i can be equal to 1 and j >= 4 to satisfy the above condition. Hence, this statement alone is not sufficient.

2. This is better, it says that p^^3 is a multiple of m, hence i must be greater than 2, and hence you can say that (p^2)(t) is a multiple of m as well.

So, the answer is B.

-Srinivas (mathguru).

I find your first explanation very interesting. Could you please explain how you got to the bold part instead of skipping the explanation.

Thax

I didn't realize , you posted another one. Very useful _________________