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

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If t, p, and q are different positive integers, how many positive inte  [#permalink]

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New post 04 Jul 2019, 07:00
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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.


 

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Re: If t, p, and q are different positive integers, how many positive inte  [#permalink]

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New post 04 Jul 2019, 08:53
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Quote:
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.


(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)
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Re: If t, p, and q are different positive integers, how many positive inte  [#permalink]

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New post 04 Jul 2019, 07:08
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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

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Re: If t, p, and q are different positive integers, how many positive inte  [#permalink]

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New post 04 Jul 2019, 08:29
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(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.
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Re: If t, p, and q are different positive integers, how many positive inte  [#permalink]

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New post 04 Jul 2019, 08:34
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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.

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Re: If t, p, and q are different positive integers, how many positive inte  [#permalink]

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New post 05 Jul 2019, 01:23
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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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Re: If t, p, and q are different positive integers, how many positive inte   [#permalink] 05 Jul 2019, 01:23
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