x+b is divisible by a

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airan: Master | Next Rank: 500 Posts; Posts: 127; Joined: Tue May 13, 2008 12:28 am; Thanked: 2 times; GMAT Score:710

x+b is divisible by a

by airan » Mon Jul 21, 2008 8:16 pm

x,a and b are positive integers. When x is divided by a, the remainder is b. When x is divided by b, the remainder is a-2. Which of the following must be true.

A. A is even.
B. x+b is divisible by a.
C. x-1 is divisible by a.
D. b=a-1
E. a+2=b+1

Thanks
Airan

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Source: Beat The GMAT — Data Sufficiency | Mon Jul 21, 2008 8:16 pm

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Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 am; Location: Montreal; Thanked: 1090 times; Followed by:355 members; GMAT Score:780

Re: x+b is divisible by a

by Ian Stewart » Tue Jul 22, 2008 4:07 am

airan wrote:x,a and b are positive integers. When x is divided by a, the remainder is b. When x is divided by b, the remainder is a-2. Which of the following must be true.

A. A is even.
B. x+b is divisible by a.
C. x-1 is divisible by a.
D. b=a-1
E. a+2=b+1

From the statements, what jumps out at me first is that a and b need to be very close in value. Recall that when you divide by, say, 7, the only possible remainders are 0, 1, 2, 3, 4, 5 and 6. That is, the remainder must be less than 7 when you divide by 7, and similarly, the remainder must be less than z when you divide by z. Onto the question:

When x is divided by a, the remainder is b.

So b < a

When x is divided by b, the remainder is a-2.

So a-2 < b.

Thus a-2 < b < a, and since b is an integer, b = a-1.

For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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maihuna: Legendary Member; Posts: 1578; Joined: Sun Dec 28, 2008 1:49 am; Thanked: 82 times; Followed by:9 members; GMAT Score:720