If x, a, and b are positive integers such that when x is

Question

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

M16-35

Bunuel · Accepted Answer

Official Solution:

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  can be expressed as  x=aq+b , where  (remainder=b) \lt (a=divisor)  (as the remainder must be less than divisor);
 
When  x  is divided by  b , the remainder is  a-2  can be expressed as  x=bp+(a-2) , where  (remainder=a-2) \lt (b = divisor) .
 
From these equations, we can deduce that:  a - 2 < b < a . Since  a  and  b  are integers, it must be true that  b = a - 1 . This is because there is only one integer between  a - 2  and  a , which is  a - 1 , and we know that this integer is equal to  b . Therefore,  b = a - 1 .

Answer: D.

Bunuel · Answer

What do you mean by "these problems"? Remainder problems or must be true problems? Anyway... Remainders: Theory: compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html Practice PS: remainder-101074.html remainder-problem-92629.html number-properties-question-from-qr-2nd-edition-ps-96030.html remainder-when-k-96127.html ps-0-to-50-inclusive-remainder-76984.html good-problem-90442.html remainder-of-89470.html number-system-60282.html remainder-problem-88102.html Practice DS: remainder-problem-101740.html remainder-101663.html ds-gcd-of-numbers-101360.html data-sufficiency-with-remainder-98529.html sum-of-remainders-99943.html ds8-93971.html need-solution-98567.html gmat-prep-ds-remainder-96366.html gmat-prep-ds-93364.html ds-from-gmatprep-96712.html remainder-problem-divisible-by-86839.html gmat-prep-2-remainder-86155.html remainder-94472.html remainder-problem-84967.html COULD or MUST be true questions: ds-number-theory-101025.html?hilit=must%20true number-properties-question-101150.html?hilit=must%20true gmat-club-please-explain-83605.html?hilit=must%20true must-be-true-101575.html?hilit=must%20true gmat-prep-question-101282.html?hilit=must%20true ab-2-c-is-even-101751.html?hilit=must%20true mgmat-inequalities-101732.html?hilit=must%20true#p788920 division-and-inequalities-87707.html?hilit=could%20true%20following#p666131 Hope it helps.

ezhilkumarank · Answer

Awesome Bunuel. Was trying hard to figure out the answer. All the information was just readily available but was looking all around.

+1 to you.

abhi758 · Answer

Thanks Bunnel!! Do you suggest using a particular strategy for these problems or using different strategy for every problem and whichever fits the bill for the given question..?

hirendhanak · Answer

Simply amazing bunuel...... you are a genius....

Mikko · Answer

cool, BUenel, i had go arround with the formulations but can not find that A-2<B tks much

suyashjhawar · Answer

Priceless info.Ton thanks

prashantbacchewar · Answer

Awesome solution Bunuel. +1

Great question Kudos to GMAT club tests

supereco87 · Answer

Awesome!!! I didnt get the fastest way, but through number plug-in I was able to figure out the solution. But your solution is so much better!!

adishail · Answer

nice and efficient !

abhi758 · Answer

Bunnel, thanks so much for the compilation! By 'these problems' I meant 'Must be true' questions in which at times you have more than 1 correct answers. My apologies for the lack of clarity their. Your compilation should be enough to practice. Thanks again!

nravi549 · Answer

Nice approach. +1 kudos

Capricorn369 · Answer

When divided by A, remainder is B, this implies A > B When divided by B, remainder is A-2, this implies B > A -2 Combining both, B < A < (B + 2) Since, A and B are integers, A = B + 1 Answer is (D) . Cheers!

rrsnathan · Answer

Way bunuel approch the problem is fantastic. I dont know whether i would be able to think the way he does . . tat too in time pressure

+1 kudos

ankit41 · Answer

Hi Bunuel,

I solved this problem with a bit different approach
x = p*a + b..........eqn(1)
and x = q*b + (a-2)..............eqn(2)

now, equating eqn(1) and eqn(2)
p*a + b = q*b + (a-2)

a(p-1) = b(q-1) - 2
if we put p = q = 3

we get, 2a = 2b - 2
or a = b - 1 
or a + 2 = b + 1 which is option E

would pl tell me where am i wrong with my approach??
Thanks.

Bunuel · Answer

You cannot assign arbitrary values to p and q and say that p = q = 3.

elizaanne · Answer

So the thing to keep in mind with remainder problems like this is that it usually involves finding a value from knowing that the remainder must be smaller than the quotient. From this, we know that a>b and that b>a-2, put these two together and we get: a>b>a-2 and since these are all positive integers then there b must equal a-2

siddharthmk · Answer

Hello Everyone,

How is option B wrong?

Bunuel · Answer

If x = 5, a = 3, and b = 2, then option B is not true: x + b = 7 is NOT divisible by a = 3.

If x, a, and b are positive integers such that when x is

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

If x, a, and b are positive integers such that when x is

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards