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To find prime p given 2p - 1 is the square of an integer

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by gmattesttaker2 » Sun Mar 30, 2014 6:52 pm
Hello,

Can you please assist with this?

If p is a prime number, what is the value of p?

(1) The sum of any p consecutive positive integers is a multiple of p.
(2) 2p - 1 is the square of an integer.

OA: E


My approach was as follows:

1) Here p could be 3 or p could be 5. Hence, in-suff.

2) Here I plugged in values till 5 and thought that this was suff. But I was wrong since p = 13 also works here. I was wondering if there is a different way to solve this problem or if for these kinds of problems we have to test at least a few values?

Thanks a lot,
Sri
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Source: — Data Sufficiency |
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by GMATGuruNY » Mon Mar 31, 2014 7:44 am
gmattesttaker2 wrote:Hello,

Can you please assist with this?

If p is a prime number, what is the value of p?

(1) The sum of any p consecutive positive integers is a multiple of p.
(2) 2p - 1 is the square of an integer.
For any set of CONSECUTIVE INTEGERS with an ODD NUMBER OF TERMS:
The SUM is always equal to a MULTIPLE OF THE NUMBER OF TERMS.
Examples:
The sum of any 3 consecutive integers is equal to a MULTIPLE OF 3.
The sum of any 5 consecutive integers is equal to a MULTIPLE OF 5.
The sum of any 13 consecutive integers is equal to a MULTIPLE OF 13.

Statement 1: The sum of any p consecutive positive integers is a multiple of p.
According to the property discussed above, p can be ANY ODD PRIME NUMBER.
INSUFFICIENT.

Statement 2: 2p - 1 is the square of an integer
Make a list of the perfect squares up to 100:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100.

Set 2p-1 equal to this list and simplify:
2p-1 = 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
2p = 2, 5, 10, 17, 26, 37, 50, 65, 82, 101.

Since 2p must be equal to an EVEN integer, the list of options can be narrowed as follows:
2p = 2, 10, 26, 50, 82
p = 1, 5, 13, 25, 41.

In the resulting list, the following options are prime:
p=5, p=13, p=41.
Since p can be different values, INSUFFICIENT.

Statements combined:
Since p=5, p=13 and p=41 satisfy both statements, the value of p cannot be determined.
INSUFFICIENT.

The correct answer is E.
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