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If \(x\), \(a\), and \(b\) are positive integers such that when \(x\) is divided by \(a\), the remainder is \(b\) and when \(x\) is divided by \(b\), the remainder is \(a-2\), then 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. \(a+2=b+1\)

If \(x\), \(a\), and \(b\) are positive integers such that when \(x\) is divided by \(a\), the remainder is \(b\) and when \(x\) is divided by \(b\), the remainder is \(a-2\), then 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. \(a+2=b+1\)

When \(x\) is divided by \(a\), the remainder is \(b\): \(x=aq+b\) and \(remainder=b \lt a=divisor\) (remainder must be less than divisor);

When \(x\) is divided by \(b\), the remainder is \(a-2\): \(x=bp+(a-2)\) and \(remainder=(a-2) \lt b=divisor\).

So we have that: \(a-2 \lt b \lt a\), as \(a\) and \(b\) are integers, then it must be true that \(b=a-1\) (there is only one integer between \(a-2\) and \(a\), which is \(a-1\) and we are told that this integer is \(b\), hence \(b=a-1\)).

aq+b=bq+a-2 factor out q's re-arranging gives: 2b=2a-2 Divide by 2 b=a-1 Ans D

Is this correct? I find it extremely confusing.

When you say "factor out q's" where does q go? It cannot just disappear.

But more importantly, the quotient should not be the same. If you check the solution above you'll see that it's x=aq+b in one case and x=bp+(a-2). We don't know whether q = p.
_________________

aq+b=bq+a-2 factor out q's re-arranging gives: 2b=2a-2 Divide by 2 b=a-1 Ans D

Is this correct? I find it extremely confusing.

When you say "factor out q's" where does q go? It cannot just disappear.

But more importantly, the quotient should not be the same. If you check the solution above you'll see that it's x=aq+b in one case and x=bp+(a-2). We don't know whether q = p.

Hi Bunuel, I am extremely sorry but I didnt understand how we got rid of the quotients and got the below equation. Can you please help me understand it

aq+b=bq+a-2 factor out q's re-arranging gives: 2b=2a-2

aq+b=bq+a-2 factor out q's re-arranging gives: 2b=2a-2 Divide by 2 b=a-1 Ans D

Is this correct? I find it extremely confusing.

When you say "factor out q's" where does q go? It cannot just disappear.

But more importantly, the quotient should not be the same. If you check the solution above you'll see that it's x=aq+b in one case and x=bp+(a-2). We don't know whether q = p.

Hi Bunuel, I am extremely sorry but I didnt understand how we got rid of the quotients and got the below equation. Can you please help me understand it

aq+b=bq+a-2 factor out q's re-arranging gives: 2b=2a-2

This is an incorrect method by gmatprepeugene2014, which is pointed out in my post.
_________________

aq+b=bq+a-2 factor out q's re-arranging gives: 2b=2a-2 Divide by 2 b=a-1 Ans D

There is a problem with approach. Its nowhere mentioned that the quotient "q" is same in both the cases as assumed here. This makes this approach incorrect.

I think this is a high-quality question and I don't agree with the explanation. Question says " when x is divided by a, the remainder is b" so that would mean that x+b is divisible by a. So the option "x+b is divisible by a" is also correct