If the sum of a set of ten different positive prime . . . .

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Vincen: Legendary Member; Posts: 2898; Joined: Thu Sep 07, 2017 2:49 pm; Thanked: 6 times; Followed by:5 members

If the sum of a set of ten different positive prime . . . .

by Vincen » Tue Jan 09, 2018 6:18 am

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If the sum of a set of ten different positive prime numbers is an even number, which of the following prime numbers CANNOT be on the set?

A. 2
B. 3
C. 5
D. 7
E. 11

The OA is the option A.

Why 2 can't be on the set? Should I try making a set of 10 prime numbers? Is there an easier way to solve this PS question?

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Source: Beat The GMAT — Problem Solving | Tue Jan 09, 2018 6:18 am

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pm; Location: Vancouver, BC; Thanked: 5254 times; Followed by:1268 members; GMAT Score:770

by Brent@GMATPrepNow » Tue Jan 09, 2018 9:04 am

Vincen wrote:If the sum of a set of ten different positive prime numbers is an even number, which of the following prime numbers CANNOT be on the set?

A. 2
B. 3
C. 5
D. 7
E. 11

Some important rules:
1. ODD +/- ODD = EVEN
2. ODD +/- EVEN = ODD
3. EVEN +/- EVEN = EVEN

4. (ODD)(ODD) = ODD
5. (ODD)(EVEN) = EVEN
6. (EVEN)(EVEN) = EVEN

The key concept here is that 2 is the only EVEN prime number. All other primes are ODD.
Since the set contains 10 different prime numbers , there are only two possible cases:
case 1) 2 is in the set of primes
case 2) 2 is NOT in the set of primes

case 1: If 2 IS in the set, then the set contains 9 ODD primes and 1 EVEN prime
In this case, the sum of the 10 primes will be ODD.
However, the question tells us that the sum is EVEN.
So, it CANNOT be the case that 2 is in the set of numbers.

Answer: A

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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regor60: Master | Next Rank: 500 Posts; Posts: 416; Joined: Thu Oct 15, 2009 11:52 am; Thanked: 27 times

by regor60 » Tue Jan 09, 2018 9:15 am

Vincen wrote:If the sum of a set of ten different positive prime numbers is an even number, which of the following prime numbers CANNOT be on the set?

A. 2
B. 3
C. 5
D. 7
E. 11

The OA is the option A.

Why 2 can't be on the set? Should I try making a set of 10 prime numbers? Is there an easier way to solve this PS question?

Prime numbers are odd numbers.

The sum of a pair of odd numbers is an even number. The sum of even numbers is an even number.

The sum of three numbers that are even, even and odd is an odd number. For example, 2, 98 and 29 = 127 is odd.

Assume the first prime is 2. That leaves 9 other primes to be added together, which is the same thing as 4 pairs of primes added to 1 additional prime.

So, adding the ten primes including 2 would be to add 2, 4 pairs of primes, and another prime.

2 is even. 4 pairs of primes is even because a pair of primes adds to an even number and even numbers added together is even. Finally, the last prime is an odd number.

Referring to the sum of three numbers rule above, the sum of these numbers is therefore odd, which contradicts the problem statement.

Therefore, 2 can't be the first prime.

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8086; Joined: Sat Apr 25, 2015 10:56 am; Location: Los Angeles, CA; Thanked: 43 times; Followed by:29 members

by Scott@TargetTestPrep » Sun Aug 04, 2019 10:29 am

Vincen wrote:If the sum of a set of ten different positive prime numbers is an even number, which of the following prime numbers CANNOT be on the set?

A. 2
B. 3
C. 5
D. 7
E. 11

Consider odd + odd = even, but odd + odd + odd = odd.

Now, if there are 10 odd numbers, the sum will be even because each pair of odds makes an even sum. But if there are 9 odd numbers and 1 even number, then the sum of the 9 odds will be odd, and adding that one even number will make the sum of those 10 numbers odd, because odd + even = odd.

Since we are told that the sum of 10 numbers is even, and since all primes are odd except 2, then 2 cannot be in the set; otherwise, the sum would be odd and not even.

Answer: A

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