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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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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, 09:53
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1
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, 08:08
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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, 08:23
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If t, p, and q are different positive integers, how many positive integers are factors of t?

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

#1
t=q^2 ; no relation of p given insufficient
#2
t=p*q and p& q have 5 + integer factors
so p =2^4 and q=3^4 so t= 16*81 factor of t ; 25 factors
IMO 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, 08:24
1
If n = a^p * b^q * c^r....., where a, b, c are prime factors.
then the number of factors of n is (p + 1)*(q + 1)*(r + 1)......

Considering statement (1) alone:
t = q^2
But we don't know if q is prime
INSUFFICIENT

Considering statement (2) alone:
t = p * q
If p and q have exactly 5 factors, then p and q must be (prime)^4
Hence, the number of factors of t = (4 + 1)(4 + 1) = 25
SUFFICIENT
The answer is (B).
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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, 08:32
If t, p, and q are different positive integers, how many positive integers are factors of t?

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

general formula for finding factors for any number of form \(a^x *b^y\)
(x+1)(y+1)
using statement 1
:
\(t = p * q\) where they have no common factor
Now p can be \(2^3\) and q can be 5^2
or p can be\(2^3 * 3^3\) and q can\(5^11 * 7*11\)
Hence we cant find a single solution

Using statement 2 :
each p and q has 5 unique factors
so \(p =3^4\) and\(q = 2^4\) is also valid therefore # number of factors = 5*5 = 25
and \(p = 2^4\)and \(q = 2^4\)is also valid therefore # number of factors = (8+1) = 9
Not sufficient
Now combine both the statements
the only valid scenario
will be of the form
\(a^x * b^y\) such as\(p =3^4\) and \(q =2^4\)

Hence total 25
Answer C
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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, 08:32
Correct answer is C.
Option 1 tells us that p and q does not have any common prime factors. But p and q can be anything. for eg: p =2 and q =3 or p=4 and q = 9.
In both cases t will have different number of factors.
Hence insufficient.

Option2:
each of p and q have 5 integral factors. But we dont know if there are some overlaps.

Hence insufficient.

Option 1 and Option2:
We know factors of both p and q. Also we know p and q does not have any common prime factor. Hence number of factors of t can be found out.

Hence sufficient.

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

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New post Updated on: 04 Jul 2019, 10:40
Statement 1: INSUFFICIENT
Case 1: p=2 & q=3, So, t=6 (4 factors of t.)
Case 2: p=4 & q=9, So, t=36 (9 factors of t.)

Statement 2: INSUFFICIENT
ONLY PERFECT SQUARES HAVE ODD NUMBER OF FACTORS.
Case 1: p=2^4 & q=2^4, So, t=2^8 [ 9 factors of t.)
Case 2:p=2^4 & q=3^4, So, t=2^4*3^4 (25 factors of t)

Combining: CASE 1 of statement 2 is NOT POSSIBLE.
ONLY CASE 1 of statement 2 is NOT POSSIBLE that means t=(any integer)^8= 9 Positive factors.

C

Originally posted by BelalHossain046 on 04 Jul 2019, 08:42.
Last edited by BelalHossain046 on 04 Jul 2019, 10:40, edited 2 times in total.
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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, 08:45
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Refer Attached Image.

Ans. 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, 08:48
(1) this means t has at least two different factors, which are p and q. However, we don't know if p or q has the value of 1. if for example, p is 1 and q is 3, then t is 3, and t has only 2 factors. If p is 2 and q is 3, then t has factors of 1, 6, 2 and 3 (four factors). So we do not know from 1.

(2) p and q have 5 positive integer factors, however we don't know if the integer factors of p and q are all different/unique numbers.

so each is not sufficient. taken together, we know that p and q are not 1, and that they each have different integer factors except for 1. hence the positive integers of t is 9. (5+5-1 as 1 will be the common integer of p and q)
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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, 08:51
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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.
No of factors of p & q are not known but it is mentioned that they have no common prime factors.
For example of p=2^3 and q =3^4, they have 4 & 5 factors respectively and no common prime factors
But if p=2 and q=3, they have 2 factors each with no common prime factors
Still it is not possible to ascertain no of positive integer factors of t based on information provided. NOT SUFFICIENT.
(2) t=p∗q; p and q each have exactly 5 positive integer factors.
If p=x^4 and q=y^4 where x is a prime number, they have both 5 positive integer factors and since p & q are different t= p*q will have 5*5 = 25 positive integer factors. SUFFICIENT.

Statement 2 alone is SUFFICIENT.

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

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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.
We do not know the number of prime factors of p and q. So, not sufficient.

(2) t=p∗q; p and q each have exactly 5 positive integer factors.
We are provided with the number of factors for p and q. Odd number of factors implies both p and q are square numbers.
But no information on common factors between the two. So not sufficient.

Taking both statements together,
Both statements together are sufficient.

Option C would be the answer.

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Originally posted by prashanths on 04 Jul 2019, 08:51.
Last edited by prashanths on 05 Jul 2019, 00:19, edited 1 time in total.
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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, 08:58
I spent 1 minute and 46 seconds. And my answer is (C).

(1) t=p∗q; p and q have no common prime factors. As we do not how many factors p or q have, there is no way to determine how many factors t has.
(2) t=p∗q; p and q each have exactly 5 positive integer factors. Here: If p and q are the same number, we can surely determine how many factors t has (no need to calculate, but the answer is 9.) But if P and q are not the same number, the factors for t will be different.

(1) and (2) together: We can tell that p and q are definitely not the same. Then, we have 2 approaches.
(a) If p and q do not share any common prime factors, they do not share other factors except for 1. That fact alone should enable us to determine that we can figure out how many factors t has.
(b) it turns out that p and q can only be expressed as primeNumber^4 (such as 16, or 27), we can determine that t has a total of 25 factors.
Sufficient.
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If t, p, and q are different positive integers, how many positive inte  [#permalink]

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New post Updated on: 04 Jul 2019, 09:15
A is the correct answer

P*q can determine the factors when co prime

B will not provide unique solutions

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Originally posted by Kssss on 04 Jul 2019, 08:58.
Last edited by Kssss on 04 Jul 2019, 09:15, edited 1 time in total.
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If t, p, and q are different positive integers, how many positive inte  [#permalink]

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New post Updated on: 05 Jul 2019, 09:51
(1) t=q∗q; p and q have no common prime factors. - means the number will be in the form q^2 but we are not aware of the factors of q and p (Just mentioned they are co prime i.e. 64 and 65) so they can have number of factors which will affect the total number of factors of the number t - Insufficient

(2) t=p∗q; p and q each have exactly 5 positive integer factors.- Provides that P and Q are perfect Squares. No information on common factors Insufficient

Combined - sufficient p=16 and Q=625 or p=625 and q=81 a^4*b^4 were a and b are prime so 25 factors

IMO C
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Originally posted by Arvind42 on 04 Jul 2019, 08:59.
Last edited by Arvind42 on 05 Jul 2019, 09:51, edited 1 time in total.
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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, 09:14
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. --> can't say how many factors of t(p=2 & q=3 or p=2*5 & q = 3*7)
(2) t=p∗q; p and q each have exactly 5 positive integer factors. --> can't say because there can be common factors between p & q

combining (1) & (2):
we can say p & q have 5 factor each & there is no common factor between them, so t have finite number of common factor.
So the answer is C
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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, 09:22
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Statement 1 is not sufficient as we do not know the value of p and q.

Statement 2 says that p and q have exactly 5 factors
Example is (x^2)^2 = x^4
Such as 2^4 = 16
Factors of 16 = 1,2,4,8,16
The same thing with 3^4 = 81
Factors of 81 = 1,3,9,27,81
Thus, this statement is sufficient, hence answer choice "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, 09:29
from:1 t=p∗q; p and q have no common prime factors.

not sufficient, since p=2,q=3 and p=2,q=9

from: 2 t=p∗q; p and q each have exactly 5 positive integer factors
Not sufficient

from both 1 and 2

Not sufficient
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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, 09:29
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1
(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, 09:30
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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.

The question needs the value of t to determine what is the number of positive integers as factors.

Stmt 1: p and q don't share any common prime factors in which t is the multiplication of p and q. but, there are many results possible for p and q. so, factors of t will also vary. hence, insufficient. Eliminate A, D.

Stmt 2: it states that t is the multiplication of p and q and also that the definite number of factors of both p and q are 5.
this is sufficient.
So, the correct answer choice is (B)
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Re: If t, p, and q are different positive integers, how many positive inte   [#permalink] 04 Jul 2019, 09:30

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