Maybe I did something wrong, or I just got lucky, so please inform me. I also had 28 as the solution, by just plugging in the values.

a. 3 / 6 = 0 r 6 INCORRECT
b. 12 / 6 = 2 r 0 INCORRECT
c. 18 / 6 = 3 r 0 INCORRECT
d. 22 / 6 = 3 r 4 & 22 / 5 = 4 r 2 INCORRECT
e. 28 / 6 = 4 r 4 & 28 / 5 = 4 r 3 CORRECT

Super-simple math, so there must be something wrong here, haha.

Bunuel – very good explanation.

First case - i). given statement in a question is - Remainder is 7 when positive integer n is divided by 18. And ii). if we are asked to find out the remainder when n is divided by 6,

Then since 18 is completely divisible by 6 or 6 is a factor of 18, we can find out the solution easily by above given statement. As, n = 18 q + 7; 18 is completely divisible by 6, thus no remainder exists when 18q/6 and when 7 is divided by 6, it would yield 1 as the remainder.

Second case i). given statement in a question is - Remainder is 7 when positive integer n is divided by 18. And ii). if we are asked to find out the remainder when n is divided by 8,

Then since 18 is not divisible by 8 or 8 is not a factor of 18, we cannot find out the solution by above given statement.

Third case i). given statement in a question is - Remainder is 7 when positive integer n is divided by 6. And ii). if we are asked to find out the remainder when n is divided by 12,

Then since 6 is not divisible by 12, however it true the other way round. We cannot find out the solution by above given statement.

Fourth case, which you have explained – “Positive integer n leaves a remainder of 4 after division by 6 and a remainder of 2 after division by 8. What is the remainder that n leaves after division by 12?”

We have two given statements – since one statement will not yield us the answer as explained in my earlier three cases. But, since we have two given statements which derive – n = 6Q1 + 4 & n = 8Q2 + 2. Q1, Q2 are quotients respectively.
And if we look at 6Q1 and 8 Q2, the LCM yields 24. Since we are asked the remainder when division is done by 12 and since 12 is completely divisible by 24. Thus, we can come up with a solution. Otherwise we cannot.

I hope I was able to explain what I wanted to and the above conclusion described in the 4 cases is correct.
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If the question were what is the remainder when n is divided by 11, then the answer would be "cannot be determined". The same if the question asked about the remainder when n is divided by 48. See, according to this general formula valid values of \(n\) are: 10, 34, 58, ... These values give different remainders upon division by 11, or 48.
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N is of the form 6p+4 which means it is one of 4, 10, 16, 22, 28, ...
N is also of the form 5q+3 which means it must also be one of 3, 8, 13, 18, 23, 28, ...

Since N must be a value in both the lists, N can take the values common to both. The first such value is 28.
So N can be 28.

Now what other values can N take? 28 is a number that will leave a remainder of 4 when divided by 6 and a remainder of 3 when divided by 5. The next such number will be (LCM of 6 and 5) + 28. Why? because whatever you add to 28, that should be divisible by 6 as well as 5. Then whatever you add will have no relevance to the remainder and the remainders will remain the same.
That is why some other values of N will be 30+28, 60+28 ...etc.

After you divide any of these numbers by 30, the remainder will be 28.

For more on divisibility and remainders, check:
http://www.veritasprep.com/blog/2011/04 ... unraveled/
http://www.veritasprep.com/blog/2011/04 ... y-applied/
http://www.veritasprep.com/blog/2011/05 ... emainders/
http://www.veritasprep.com/blog/2011/05 ... s-part-ii/

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?
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No. It is 650-700 level question. It seems quite do-able only because we have done the concept of remainders in great detail. For a newbie, this problem can be quite challenging.
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In this case we wouldn't be able to determine the remainder. For example, if we were asked to find the remainder of n divided by 11, then we would get different answers: if n = 28, then remainder would be 6 but if n = 58, then the remainder would be 3.
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You might want to check out this post first on divisibility and remainders:
http://www.veritasprep.com/blog/2011/04 ... unraveled/

This tells you why when you divide a number by 30 and get, say 4 as remainder, you will get 4 as remainder when you divide the same number by 6 or 5 or 10 or 15 (factors of 30 greater than 4).

After that, you should check out this post which discusses this particular question in detail:
http://www.veritasprep.com/blog/2014/05 ... -the-gmat/

I think these two together will help you get a very clear picture.
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N= 6x + 4 --1
N= 5y + 3 --2
from 1 and 2
6x+1 = 5y
this seems simple now , a multiple of 6 to which when 1 is added it turns into a multiple of 5 ; 24+1=25

x=4, N=28 ; 28%30 = 28 .

To understand the 'why' of divisibility and remainders, check out these posts in this order:

http://www.veritasprep.com/blog/2011/04 ... unraveled/
http://www.veritasprep.com/blog/2011/04 ... y-applied/
http://www.veritasprep.com/blog/2011/05 ... emainders/
http://www.veritasprep.com/blog/2011/05 ... s-part-ii/
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Did you check out the last link provided above? Here it is again: http://www.veritasprep.com/blog/2011/05 ... s-part-ii/

With a case such as this: x = 8n + 3 = 6m + 4, for the first number of this form, you need to use hit and trial.
Note that this example is incorrect since 8n+3 is an odd number and 6m+4 is an even number. But check the post to see how to apply the method to correct questions.
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