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04 Jul 2019, 08:00
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B
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Difficulty:
95%
(hard)
Question Stats:
34% (01:44) correct
66%
(01:40)
wrong
based on 398
sessions
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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.
(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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SageJiraiya
Joined: 14 Jan 2016
Last visit: 30 Jul 2022
Posts: 108
Given Kudos: 84
Location: India
Concentration: Marketing, General Management
Schools: Kenan-Flagler '22 (A$) Jones '22 (A$) Kelley '22 (A$) Mendoza '22 (D) Tepper '22 (WL) Marshall '22 (D) Duke '22 (WA$) Johnson '22 (WL) Darden '22 (D) Ross '22 (WA$) Wharton '23 (D)
GMAT 1: 710 Q50 V35
GMAT 2: 750 Q50 V41
WE:Marketing (Advertising and PR)
Schools: Kenan-Flagler '22 (A$) Jones '22 (A$) Kelley '22 (A$) Mendoza '22 (D) Tepper '22 (WL) Marshall '22 (D) Duke '22 (WA$) Johnson '22 (WL) Darden '22 (D) Ross '22 (WA$) Wharton '23 (D)
GMAT 2: 750 Q50 V41
Posts: 108
04 Jul 2019, 09:53
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Quote:
(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)
General Discussion
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
C
(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
