If t, p, and q are different positive integers, how many positive inte

Question

If t, p, and q are different positive integers, how many positive integers are factors of t?

(1)  t = p*q ; p and q have no common prime factors.
(2)  t = p*q ; p and q each have exactly 5 positive integer factors.

SageJiraiya · Accepted Answer

(1) t=p∗q;  p and q have no common prime factors.
We can't arrive to the number of factors of q with this information.
p can be 2 and q can be 3.
This would make t = 6, number of factors would be 4 (1,2,3 and 6)
Or p can be 4 and q can be 3
This would make t = 12, number of factors would be 6 (1,2,3,4,6 and 12)
 Insufficient.

(2) t=p∗q;  p and q each have exactly 5 positive integer factors.
Let N be a number. N =  a^x b^y c^z  , where a, b and c are different prime numbers.
The number of factors of N =  (x+1)*(y+1)*(z+1) 
Now, for a positive number to have odd number of factors, it needs to be a perfect square.
since both p and q have 5 factors and both are  different (given in stem) , they must be fourth powers of different primes.
let  p = a^4  and  q = b^4  where both a and b are distinct primes.
Now  t = p*q = a^4 * b^4 
So the number of factors t will have = (4+1) * (4+1) = 25 
 Sufficient.

Hence  (B)

umarovaalbina · Answer

If t, p, and q are different positive integers, how many positive integers are factors of t?

(1) t=q∗q; p and q have no common prime factors.
This statement doesn't tell us about the number of factors of q (we don't know if q is prime)
For example, it could be that t = 12*12 or t = 13*13 etc

(2) t=p∗q; p and q each have exactly 5 positive integer factors.

We don't know if these integer factors are the same for p and q. They may be different or the same

However, (1) and (2) combined allow us to understand that those prime factors for p and q are different
Sufficient to answer the question

C

QuantMadeEasy · Answer

(1) t=p∗q; p and q have no common prime factors.
p and q are coprimes, let p=2, q=3 then no. of factors of t are 4
When p=6 and q=7; no. of factors of t are 8. Insufficient.

(2) t=p∗q; p and q each have exactly 5 positive integer factors.
As p and q are different positive integers and each have 5 factors, then number of factors of t are 25. Sufficient.

B is correct.

RajatVerma1392 · Answer

Hi,

1. t = p * q where p and q have no common prime factors.
That means they are relatively prime or co-prime numbers.
let p = 40 =>  2^3  * 5
and q = 21 => 3 * 7 
thus t =  2^3  * 3 * 5 *7
so, t can have 32 factors.
Sufficient.

2. t = p * q where p and q have exactly 5 positive integer factors. 
Only the SQUARES of integers have an ODD number of positive factors.
so, let p = 16 =>  2^4  and q = 81 => 3^4 
so, t =  2^4 *  3^4 
total factor = 25
Sufficient.

The answer is D.

Please hit kudos if you like the solution.

ccheryn · Answer

If t, p, and q are different positive integers, ( note the word different)

(1) t=p∗qt=p∗q; p and q have no common prime factors. clearly insufficient 
(2) t=p∗qt=p∗q; p and q each have exactly 5 positive integer factors. so p and q are perfect squares, it give rise to two possibilities either p = q or p not equal to q, question says they are different

so different perfect squares results in x^4 * y^4 hence total factors is 5 * 5 = 25 for t so b alone is sufficient

ans B

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If t, p, and q are different positive integers, how many positive inte

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