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25 May 2011, 11:18
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1. Positive integer n leaves a remainder of 4 after division by 6 and a remainder of 3 after division by 5. If n is greater than 30, what is the remainder that n leaves after division by 30?
(A) 3
(B) 12
(C) 18
(D) 22
(E) 28
Solution:
Positive integer n leaves a remainder of 4 after division by 6 --> --> 4, 10, 16, 22, 28, ...
Positive integer n leaves a remainder of 3 after division by 5 --> --> 3, 8, 13, 18, 23, 28, ...
- we have 30 as lcm of 5 and 6 is 30 and we have 28 as the first common integer in the above patterns is 28.
Hence remainder when positive integer n is divided by 30 is 28.
Answer: E.
2. QR. 68. When positive integer n is divided by 5, the remainder is 1. When n is divided by 7, the remainder is 3. What is the smallest positive integer k such that k+n is a multiple of 35?
a) 3
b) 4
c) 12
d) 32
e) 35
Solution:
Positive integer n is divided by 5, the remainder is 1 --> , where is the quotient --> 1, 6, 11, 16, 21, 26, 31, ...
Positive integer n is divided by 7, the remainder is 3 --> , where is the quotient --> 3, 10, 17, 24, 31, ....
You can not use the same variable for quotients in both formulas, because quotient may not be the same upon division n by two different numbers.
For example 31/5, quotient q=6 but 31/7, quotient p=4.
Ans. B
What is the difference between the requirements of these two problems?
(A) 3
(B) 12
(C) 18
(D) 22
(E) 28
Solution:
Positive integer n leaves a remainder of 4 after division by 6 --> --> 4, 10, 16, 22, 28, ...
Positive integer n leaves a remainder of 3 after division by 5 --> --> 3, 8, 13, 18, 23, 28, ...
- we have 30 as lcm of 5 and 6 is 30 and we have 28 as the first common integer in the above patterns is 28.
Hence remainder when positive integer n is divided by 30 is 28.
Answer: E.
2. QR. 68. When positive integer n is divided by 5, the remainder is 1. When n is divided by 7, the remainder is 3. What is the smallest positive integer k such that k+n is a multiple of 35?
a) 3
b) 4
c) 12
d) 32
e) 35
Solution:
Positive integer n is divided by 5, the remainder is 1 --> , where is the quotient --> 1, 6, 11, 16, 21, 26, 31, ...
Positive integer n is divided by 7, the remainder is 3 --> , where is the quotient --> 3, 10, 17, 24, 31, ....
You can not use the same variable for quotients in both formulas, because quotient may not be the same upon division n by two different numbers.
For example 31/5, quotient q=6 but 31/7, quotient p=4.
Ans. B
What is the difference between the requirements of these two problems?
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25 May 2011, 18:10
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I am not sure I understand your question very well but I am assuming that what is troubling you is the quotient you get upon division. Well, quotient is not our worry in either of the questions. We only need to focus on the remainders.
Both the questions use the same concept. They just ask you for the answer in different formats.
Q1
Positive integer n leaves a remainder of 4 after division by 6 --> --> 4, 10, 16, 22, 28, ...
Positive integer n leaves a remainder of 3 after division by 5 --> --> 3, 8, 13, 18, 23, 28, ...
n satisfies both the conditions. The first such number is 28. So you get 28 as the smallest value of n. Next value of n will be 30+28 = 58 since 30 is the LCM of 6 and 5. Next will be 30+58 = 88 and so on
what is the remainder that n leaves after division by 30
When you divide 28/58/88... by 30, you get 28 as remainder.
Q2
Positive integer n is divided by 5, the remainder is 1 --> , where is the quotient --> 1, 6, 11, 16, 21, 26, 31, ...
Positive integer n is divided by 7, the remainder is 3 --> , where is the quotient --> 3, 10, 17, 24, 31, ....
n satisfies both the conditions. The first such number is 31. Hence smallest value of n is 31. Next value is 31+35 = 66 (35 is the LCM of 5 and 7) ... and so on
What is the smallest positive integer k such that k+n is a multiple of 35?
So n = 31/66... Is n a multiple of 35? No. 31 is not divisible by 35. What should you add to 31 to make it divisible by 35? The smallest number you could add is 4 because 31+4 = 35 which is divisible by 35.
So answer is 4.
I have been discussing Divisibility and Remainders for the past few weeks on my blog. The following posts might help you:
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/0 ... unraveled/
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/0 ... y-applied/
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/0 ... emainders/
Both the questions use the same concept. They just ask you for the answer in different formats.
Q1
Positive integer n leaves a remainder of 4 after division by 6 --> --> 4, 10, 16, 22, 28, ...
Positive integer n leaves a remainder of 3 after division by 5 --> --> 3, 8, 13, 18, 23, 28, ...
n satisfies both the conditions. The first such number is 28. So you get 28 as the smallest value of n. Next value of n will be 30+28 = 58 since 30 is the LCM of 6 and 5. Next will be 30+58 = 88 and so on
what is the remainder that n leaves after division by 30
When you divide 28/58/88... by 30, you get 28 as remainder.
Q2
Positive integer n is divided by 5, the remainder is 1 --> , where is the quotient --> 1, 6, 11, 16, 21, 26, 31, ...
Positive integer n is divided by 7, the remainder is 3 --> , where is the quotient --> 3, 10, 17, 24, 31, ....
n satisfies both the conditions. The first such number is 31. Hence smallest value of n is 31. Next value is 31+35 = 66 (35 is the LCM of 5 and 7) ... and so on
What is the smallest positive integer k such that k+n is a multiple of 35?
So n = 31/66... Is n a multiple of 35? No. 31 is not divisible by 35. What should you add to 31 to make it divisible by 35? The smallest number you could add is 4 because 31+4 = 35 which is divisible by 35.
So answer is 4.
I have been discussing Divisibility and Remainders for the past few weeks on my blog. The following posts might help you:
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/0 ... unraveled/
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/0 ... y-applied/
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/0 ... emainders/
Archived Topic
Hi there,
This topic has been closed and archived due to inactivity or violation of community quality standards. No more replies are possible here.
Where to now? Join ongoing discussions on thousands of quality questions in our Quantitative Questions Forum
Still interested in this question? Check out the "Best Topics" block above for a better discussion on this exact question, as well as several more related questions.
Thank you for understanding, and happy exploring!
