We firmly believe that teaching someone is a most productive learning for oneself and every now and then, something happens that strengthens this belief of ours. It’s the questions people ask – knowingly or unknowingly – that connect strings in our mind such that we feel we have gained more from the discussion than even our students!

The other day, we came across this common GMAT question on remainders and many people had solved it the way we would expect them to solve. One person, perhaps erroneously, used a shortcut which upon reflection made perfect sense. Let me give you that question and the shortcut and the problem with the shortcut. We would like you to reflect on why the shortcut actually does make sense and is worth noting down in your log book.

Question: 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: We are assuming you know how people do the question usually:

The logic it uses is discussed here and the solution is given below as Method I.

Method I:

Positive integer n leaves a remainder of 4 after division by 6. So n = 6a + 4

n can take various values depending on the values of a (which can be any non negative integer).

Some values n can take are: 4, 10, 16, 22, 28, …

Positive integer n leaves a remainder of 3 after division by 5. So n = 5b + 3

n can take various values depending on the values of a (which can be any non negative integer).

Some values n can take are: 3, 8, 13, 18, 23, 28, …

The first common value is 28. So n = 30k + 28

Hence remainder when positive integer n is divided by 30 is 28.

Perfect! But one fine gentleman came up with the following solution wondering whether he had made a mistake since it seemed to be “super simple Math”.

Method II:

Given in question: “n leaves a remainder of 4 after division by 6 and a remainder of 3 after division by 5.”

Divide the options by 6 and 5. The one that gives a remainder of 4 and 3 respectively will be correct.

(A) 3 / 6 gives Remainder = 3 -> INCORRECT
(B) 12 / 6 gives Remainder = 0 -> INCORRECT
(C) 18 / 6 gives Remainder = 0 -> INCORRECT
(D) 22 / 6 gives Remainder = 4 but 22 / 5 gives  Remainder = 2 -> INCORRECT
(E) 28 / 6 gives Remainder = 4 and 28 / 5 gives Remainder = 3 ->  CORRECT

Now let us point out that the options are not the values of n; they are the values of remainder that is leftover after you divide n by 30. The question says that n must give a remainder of 4 upon division by 6 and a remainder of 3 upon division by 5. This solution divided the options (which are not the values of n) by 6 and 5 and got the remainder as 4 and 3 respectively. So the premise that when you divide the correct option by 6 and 5, you should get a remainder of 4 and 3 respectively is faulty, right?

This is where we want you to take a moment and think: Is this premise actually faulty?

The fun part is that method II is perfectly correct too. Method I seems a little complicated when compared with Method II, doesn’t it? Let us give you the logic of why method II is correct:

Recall that division is nothing but grouping. When you divide n by 30, you make complete groups of 30 each. The number of groups you get is called the quotient (not relevant here) and the leftover is called the remainder. If this is not clear, check this post first.

When n is divided by 30, groups of 30 are made. Whatever is leftover is given in the options. 30 is completely divisible by 6 and by 5 hence the groups of 30 can be evenly divided into groups of 6 as well as groups of 5. Now, whatever is leftover (given in the options) after division by 30, we need to split that into further groups of 6 and 5. When we split it into groups of 6 (i.e. divide the option by 6), we must have remainder 4 since n leaves remainder 4. When we split it into groups of 5 (i.e. divide the option by 5), we must have remainder 3 since n leaves remainder 3. And, that is the reason we can divide the options by 6 and 5, check their remainders and get the answer!

Now, isn’t that neat!

Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!

### 4 Responses

1. Maria says:

Hi!
I just dont understand why the first common value (28) has to be the remainder.. May you explain me this part please?
Thanks you!

• Karishma says:

The first common value is the first value that n can take. It is of the form 6a+4 as well as 5b+3. This value is 28.
28 = 6 + 6 + 6 + 6 + 4
28 = 5 + 5 + 5 + 5 + 5 + 3

Now, which is the next value which will be of the form 6a+4 as well as 5b+3?
I still want the remainders to remain at 4 and 3. What can I add to 28? If I add another 6, remainder when divided by 5 will change. If instead I add 5, the remainder when divided by 6 will change. What about their LCM? If I add 30 (which is sum of five 6s or sum of six 5s) to it and get 58,

28 = (6 + 6 + 6 + 6 + 6) + 6 + 6 + 6 + 6 + 4
28 = (5 + 5 + 5 + 5 + 5 + 5) + 5 + 5 + 5 + 5 + 5 + 3

Similarly, we can add any multiple of 30. So we get n = 30k + 28 where k could be any non negative integer.

2. Frank says:

Is it possible to just recognize the difference of 2 between the #s for n/6 = r4 (6-4=2) and n/5=r3 (5-3=2) or is that just a fluke that they happened to be differences of 2 because I applied that to see that what is a difference of 2 for 30, and 28 was on the board.

Just curious if that logic is applied or even reasonable.

Thanks

• Karishma says:

Yes, you are absolutely correct. The two have a common negative remainder i.e. 2 so you will get a remainder of 30 – 2 = 28 when you divide by 30.