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A "gamma sequence" is defined as an infinite sequence of positive integers where no integer appears more than once, and there is a finite number of prime numbers in that sequence. The sequence H in an infinite sequence of positive integers, where no integer appears more than once. Is H a gamma sequence?
1. There are infinite many multiples of 4 in H.
2. Only the first 30 integers in the sequence H are odd, and there is at least one prime integer in H.
Source: Kaplan
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1. There are infinite many multiples of 4 in H.
2. Only the first 30 integers in the sequence H are odd, and there is at least one prime integer in H.
Source: Kaplan
Posted from my mobile device
27 Aug 2020, 09:15
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Laila12618
A "gamma sequence" is defined as an infinite sequence of positive integers where no integer appears more than once, and there is a finite number of prime numbers in that sequence. The sequence H in an infinite sequence of positive integers, where no integer appears more than once. Is H a gamma sequence?
1. There are infinite many multiples of 4 in H.
2. Only the first 30 integers in the sequence H are odd, and there is at least one prime integer in H.
1. There are infinite many multiples of 4 in H.
2. Only the first 30 integers in the sequence H are odd, and there is at least one prime integer in H.
Using Statement 1, the sequence could just be 4, 8, 12, 16, ..., etc, and there might be no primes at all in the sequence. So it could be a gamma sequence. But it could also alternate between multiples of 4 and prime numbers - the sequence could be something like 4, 2, 8, 3, 12, 5, 16, 7, 20, 11, ... in which case it contains an infinite number of multiples of 4, but also an infinite number of primes. Then it's not a gamma sequence. So Statement 1 is not sufficient.
Using Statement 2, we know there are only 30 odd numbers in the sequence. So there can be at most 30 odd primes in the sequence. There is only one even prime, namely '2', and since every number in the sequence is different, we could only have at most one '2' in the sequence among the even numbers after the first thirty terms. So from Statement 2, we have no more than 31 prime numbers in the sequence, and certainly have a finite number of primes, and Statement 2 is sufficient.
So the answer is B.
31 Aug 2020, 08:26
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IanStewart
Laila12618
A "gamma sequence" is defined as an infinite sequence of positive integers where no integer appears more than once, and there is a finite number of prime numbers in that sequence. The sequence H in an infinite sequence of positive integers, where no integer appears more than once. Is H a gamma sequence?
1. There are infinite many multiples of 4 in H.
2. Only the first 30 integers in the sequence H are odd, and there is at least one prime integer in H.
1. There are infinite many multiples of 4 in H.
2. Only the first 30 integers in the sequence H are odd, and there is at least one prime integer in H.
Using Statement 1, the sequence could just be 4, 8, 12, 16, ..., etc, and there might be no primes at all in the sequence. So it could be a gamma sequence. But it could also alternate between multiples of 4 and prime numbers - the sequence could be something like 4, 2, 8, 3, 12, 5, 16, 7, 20, 11, ... in which case it contains an infinite number of multiples of 4, but also an infinite number of primes. Then it's not a gamma sequence. So Statement 1 is not sufficient.
Using Statement 2, we know there are only 30 odd numbers in the sequence. So there can be at most 30 odd primes in the sequence. There is only one even prime, namely '2', and since every number in the sequence is different, we could only have at most one '2' in the sequence among the even numbers after the first thirty terms. So from Statement 2, we have no more than 31 prime numbers in the sequence, and certainly have a finite number of primes, and Statement 2 is sufficient.
So the answer is B.
Hi IanStewart,
Thanks for the explanation. I have one doubt though. The stem also describes the sequence as one having positive integers only. How does statement B satisfy the +ve integer condition of the sequence? So, the 1st 30 numbers may very well be -ve odd integers with the 31st one being "2".
Pls let me know if I am understanding this correctly.
Thanks in advance.
VJ
