If the sum of two prime numbers x and y is odd,

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VJesus12: Legendary Member; Posts: 2276; Joined: Sat Oct 14, 2017 6:10 amFollowed by: 3 members

by VJesus12 » Sat Oct 14, 2017 6:42 am

If the sum of two prime numbers x and y is odd, then the product of x and y must be divisible by

(A) 2
(B) 3
(C) 4
(D) 5
(E) 8

The OA is A.

I got confused here. Can be the sum of two odd numbers and odd number? the prime numbers are all odd. I really need some help here please.

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Source: Beat The GMAT — Problem Solving | Sat Oct 14, 2017 6:42 am

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

by Brent@GMATPrepNow » Sat Oct 14, 2017 6:49 am

VJesus12 wrote:If the sum of two prime numbers x and y is odd, then the product of x and y must be divisible by

(A) 2
(B) 3
(C) 4
(D) 5
(E) 8

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

If x + y = ODD, then one of the values must be ODD and the other value must be EVEN
Since 2 is the ONLY EVEN prime number, then we can be certain that either x or y equals 2.
So, xy = (2)(some ODD prime number)
In other words, xy = (EVEN )(ODD) = EVEN (from rule 5)
So, xy is EVEN, which means xy is divisible by 2
Answer: A

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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EconomistGMATTutor: GMAT Instructor; Posts: 555; Joined: Wed Oct 04, 2017 4:18 pmFollowed by: 12 members; Thanked: 180 times

by EconomistGMATTutor » Sat Oct 14, 2017 7:25 am

Hello Vjesus12.

First of all, not all the prime numbers are odd numbers. The number 2 is the only prime number which is an even number.

Now, if we add two odd numbers the result will be an even number.

This is not our situation. We have the result of the sum is an odd number. The only way to get this is adding an even number plus and odd number.

So, one of the prime numbers is even. It implies one of the prime number is 2. Let's say x=2.

Now, we have that x*y=2*y. So the product is an even number.

In conclusion the product of x by y is divisible by 2.

The correct answer is A.

I hope it helps you.

I'm available if you'd like any follow up.

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by Brent@GMATPrepNow » Sat Oct 14, 2017 7:36 am

VJesus12 wrote:If the sum of two prime numbers x and y is odd, then the product of x and y must be divisible by

(A) 2
(B) 3
(C) 4
(D) 5
(E) 8

Alternatively, we can use the process of elimination to solve the question.

The question asks "the product of x and y must be divisible by..."
So, if we can show that the product xy is NOT divisible by a certain answer choice, then we can ELIMINATE that answer choice.

GIVEN: the sum of two prime numbers x and y is odd
So, for example, x = 2 and y = 7 satisfies the given information.
xy = (2)(7) = 14, so it's possible that xy = 14

Now check the answer choices....

(A) 2. 14 IS divisible by 2. So, KEEP A for now.
(B) 3. 14 is NOT divisible by 3. ELIMINATE B
(C) 4. 14 is NOT divisible by 4. ELIMINATE C
(D) 5. 14 is NOT divisible by 5. ELIMINATE D
(E) 8. 14 is NOT divisible by 8. ELIMINATE E

By the process of elimination, the correct answer is A

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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by [email protected] » Sat Oct 14, 2017 11:03 am

Hi VJesus12,

This question is all about Number Property rules, so you can either TEST VALUES or use your knowledge of those NP rules to answer the question. As an aside, the fact that X and Y are PRIME is actually irrelevant to this question. If the two variables are positive integers, and their sum is ODD, then one of the variables MUST be EVEN and the other MUST be ODD. The product of an even and an odd is ALWAYS EVEN. By extension, even numbers are ALWAYS divisible by 2.

Final Answer: A

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Rich

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EconomistGMATTutor: GMAT Instructor; Posts: 555; Joined: Wed Oct 04, 2017 4:18 pmFollowed by: 12 members; Thanked: 180 times

by EconomistGMATTutor » Sun Oct 15, 2017 2:23 pm

Hi VJesus12,
Let's take a look at your question.

The question states that the sum of two prime numbers x and y is odd.
Sum of two numbers is odd only when one number is even and the other is odd.

SUM (ODD) = EVEN NUMBER + ODD NUMBER

Now, we will see if the product of an even number and an odd number is even or odd?
(EVEN NUMBER)(ODD NUMBER) = ?

For example, even number = 8 and odd number = 3
then product = 8(3) = 24, which is an even number.

Therefore,
(EVEN NUMBER)(ODD NUMBER) = EVEN NUMBER

And an even number is always divisible by 2.
Therefore, Option A is correct.

Hope this helps.

I am available, if you'd like any follow up.

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Jeff@TargetTestPrep: GMAT Instructor; Posts: 1462; Joined: Thu Apr 09, 2015 9:34 amFollowed by: 22 members; Location: New York, NY; Thanked: 39 times

by Jeff@TargetTestPrep » Mon Dec 18, 2017 9:16 am

VJesus12 wrote:If the sum of two prime numbers x and y is odd, then the product of x and y must be divisible by

(A) 2
(B) 3
(C) 4
(D) 5
(E) 8

All prime numbers are odd, except for 2. Recall that odd + odd = even and that even + odd = odd. If the sum of two prime numbers is odd, then we see that one of the numbers must be 2 and the other number must be odd. Thus, the product xy is a multiple of 2 and must be divisible by 2.

Answer: A

Jeffrey Miller
Head of GMAT Instruction
[email protected]

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