Another Remainder problem.

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vaibhavjha: Newbie | Next Rank: 10 Posts; Posts: 6; Joined: Thu Aug 19, 2010 7:26 am; Thanked: 1 times

by vaibhavjha » Fri Nov 12, 2010 12:54 pm

Q) Is P-Q divisible by 45?
(1) When positive integer P is divided by 9, the remainder is 2; and when P is divided by 5 the remainder is 1
(2) When positive integer Q is divided by 9, the remainder is 2; and when Q is divided by 5 the remainder is 1

OA[spoiler]:C[/spoiler]

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Source: Beat The GMAT — Data Sufficiency | Fri Nov 12, 2010 12:54 pm

beat_gmat_09: Master | Next Rank: 500 Posts; Posts: 437; Joined: Sat Nov 22, 2008 5:06 amFollowed by: 1 member; Location: India; Thanked: 50 times; GMAT Score:580

by beat_gmat_09 » Fri Nov 12, 2010 6:53 pm

vaibhavjha wrote:Q) Is P-Q divisible by 45?
(1) When positive integer P is divided by 9, the remainder is 2; and when P is divided by 5 the remainder is 1
(2) When positive integer Q is divided by 9, the remainder is 2; and when Q is divided by 5 the remainder is 1

OA[spoiler]:C[/spoiler]

stmt 1 - P/9 remainder 2, P/5 remainder 1; pick number 11, satisfies both the conditions, no additional info, P-Q cannot be know using this info.

stmt 2 - Q/9 remainder 2, P/5 remainder 1; pick number 11 or 56 or 144, satisfies both the conditions, no additional info, P-Q cannot be know using this info.

Combine 1 & 2 - if P - 11 Q - 11 or 56 or 126 and so on .. difference 11-11 = 0 div by 45, 11-56 = 45 div by 45, 11-144 = 135 divisible by 45.
Both are sufficient.

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Rahul@gurome: GMAT Instructor; Posts: 1179; Joined: Sun Apr 11, 2010 9:07 pmFollowed by: 88 members; Location: Milpitas, CA; Thanked: 447 times

by Rahul@gurome » Fri Nov 12, 2010 8:48 pm

From (1) alone, we cannot say anything because nothing is being said about Q.
Similarly, from (2) alone we cannot say anything because nothing is being said about P.
We next combine both the statements together and check.
From (1) we have P = 9a + 2 and P = 5b + 1 where a and b are integers.
From (2) we have Q = 9c + 2 and Q = 5d + 1 where c and d are integers.

Taking the corresponding differences we get that P - Q = 9(a - c) and also P - Q = 5(b - d).
Since (a-c) and (b - d) are integers, (P - Q) is divisible by both 9 and 5.
Hence, since 9 and 5 are co prime, we conclude that P - Q is divisible by 9 * 5 = 45.
So both statements together are sufficient to answer the question.
The correct answer is (C).

Rahul Lakhani
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