Lesson: Introduction to Remainders

Comment on Introduction to Remainders

Can you help me with this?

When positive integer A is divided by positive integer B, the result is 9.35. Which of the following could be the remainder when A is divided by B ?

(A) 13
(B) 14
(C) 15
(D) 16
(E) 17
gmat-admin's picture

0.35 = 35/100
So, we can write: A/B = 9 35/100 = 935/100
So, one possible case is that A = 935 and B = 100
When we divide 935 by 100, the remainder is 35
Unfortunately 35 is NOT one of the answer choices.

Now let's examine some fractions that are EQUIVALENT to 935/100
For example, 935/100 = 187/20 (= 9.35)
In other words, A = 187 and B = 20
In this case, when we divide A by B, the remainder is 7
Unfortunately 7 is NOT one of the answer choices.

Find another fraction EQUIVALENT to 935/100
How about 374/40 (= 9.35)
In other words, A = 374 and B = 40
In this case, when we divide A by B, the remainder is 14
Aha - 14 IS one of the answer choices.

Answer: B

Hi Brent, is there a generalized relationship between the decimal number and the remainder so that we can solve this type of question quickly? Or the only way is by trials and errors as shown in your solution? Thanks
gmat-admin's picture

I don't believe there's a relationship that would be of much use. The reason for this is that everything depends on the value of the dividend.

Hi Brent, would we get such questions on the GMAT? If so, what is the difficulty level of this question?
gmat-admin's picture

Yes, that type of question would be within the score of the GMAT. I'd say it's a 650-level question.

Hey Brent, I'm quite confused with two major takeaways from this video.

My first takeaway is that in the equation N/D = Q(R)...then the possible values of N are R, R+D, R+2D, ..R+5D and so on.

My second takeaway is the formula which states N=(QxD)+R.
However, doing practice questions at the 350-500 level, I find that the second takeaway/the formula does work because I have two unknowns: the Dividend(N) and the Quotient.

Whereas, in the first takeaway, the only information I need to know is the Divisor and the Remainder in order to find N.

For example: If the remainder is 7 when positive integer n is divided by 18, what is the remainder when n is divided by 6?

If I use the equation method of N=(QxD)+R I get N = 18Q+7 and N = 6Q+R. This leaves me with too many variables.

However, given my first takeaway. I can infer that if N=R or N=R+D or N=R+2D. I can say that N is 7, 25 or 43 (or any number following this rule). As a result, testing 7/6 gives me R1 which gives me the correct solution.

As a result, I would like to clarify/differentiate between these two methods presented in the video because I can't seem to comprehend when to use the formula N=(QxD)+R.

Thank you.
gmat-admin's picture

Here's where you got into trouble:

"For example: If the remainder is 7 when positive integer n is divided by 18, what is the remainder when n is divided by 6?
If I use the equation method of N=(QxD)+R I get N = 18Q+7 and N = 6Q+R. This leaves me with too many variables."

Your first equation, N = 18Q+7, is perfect, however your second equation, N = 6Q+R, is not correct, because we can't use the variable Q a second time.
Notice that Q, the quotient (when dividing by 18) in your first equation, will not be same as the quotient when dividing by 6.

So, we need to use a different variable for the second equation. For example, we might say that N = 6k + R, where k represents the quotient when dividing by 6.

This, of course, means EVEN MORE variables!! So, at this point, we must either try to keep going somewhere with our 2 equations or try something new.
I suggest we try something new (as you did).

Conversely, we might try to take our first equation N = 18Q + 7, and try to see if this (alone) will help us determine the remainder when N is divided by 6.

To do so, we can rewrite the equation N = 18Q + 7 as follows:
Take: N = 18Q + 7
Rewrite 18Q as (6)(3Q) to get: N = (6)(3Q) + 7
Rewrite 7 as 6 + 1 to get: N = (6)(3Q) + 6 + 1
Now factor out a 6 from the first two terms to get: N = 6(3Q + 1) + 1

KEY CONCEPT: Notice that 6(3Q + 1) represents some multiple of 6.
So, 6(3Q + 1) + 1 is ONE GREATER than some multiple of 6.
So, if we divide 6(3Q + 1) + 1 by 6, the remainder will be 1.
In other words, if we divide N by 6, the remainder will be 1.

Does that help?



Sir please explain
gmat-admin's picture

using statement 1&2
n= 13a+b

remainder of n/26 will be remainder of 13a+b/26
13a/26= a/2 remainder will be 1 but we divided the expression by 13 so remainder will be 1*13 = 13
b = 3

3/26 remainder = 3
so remainder is 13+3= 16
is this approach correct???
gmat-admin's picture

Yes, that approach is correct

For the positive integers q, r, s, and t, the remainder when q is divided by r is 7 and the remainder when s is divided by t is 3. All of the following are possible values for the product rt EXCEPT

gmat-admin's picture

Hi Brent,

wondering if there's a quick way to solve this question: https://gmatclub.com/forum/if-n-775-778-781-what-is-the-remainder-when-n-is-divided-by-217323.html
gmat-admin's picture

Hi Brent,

I tried to use the previous question logic to answer this question but couldn't get it.

gmat-admin's picture

Did you mean to post the same link that I posted?


Oh no, didn't realize I posted the same link. My apologies. Here's the one I was referring to.

gmat-admin's picture

Question link: https://gmatclub.com/forum/if-x-2891-2892-2893-2898-2899-2900-the-164102...

This question and the previous question are both skirting the boundaries of out-of-scope-land.

To get a reasonably fast solution for this latest question, we need to apply the rules of modular arithmetic, which is not required for the GMAT.

That said, the solution to https://gmatclub.com/forum/if-x-2891-2892-2893-2898-2899-2900-the-164102... is pretty much identical to the solution to https://gmatclub.com/forum/if-n-775-778-781-what-is-the-remainder-when-n...

If you show me your attempted solution to this latest question, I'll be happy to indicate where the issue lies.



I don't think that there's any value for x whereby 8 is the remainder when divided by 20. Am I right to assume then that we only consider remainders without simplifying the fraction? E.g.

28 / 20 = 1*8/20 the remainder being 8 and not 2
gmat-admin's picture

Question link: https://gmatclub.com/forum/when-positive-integer-x-is-divided-by-20-the-...

Be careful, 28/20 ≠ 1*8/20
28/20 = 1 + 8/20

You're right though, when determining the remainder, we don't reduce the fraction 8/20 to 2/5



Would testing numbers be a valid approach here? Or is this meant to catch people out?
gmat-admin's picture

Question link: https://gmatclub.com/forum/the-remainder-when-the-positive-integer-m-is-...

Yes, we can certainly test values.
If you check some of the other posts on that thread, you'll find students who successfully answered the question with that approach.


Hi Brent,

question link https://gmatclub.com/forum/when-a-positive-integer-n-is-divided-by-7-what-is-the-remainder-226317.html

Can you, please check my solution:

Statement 1) (n-294)/7 = Q+3 (where Q is some quotient Q), it is same as n/7 - 294/7 = Q+3, same as n/7 - 42 = Q+3 ==> n/7 = Q+45, hence remainder is 45.

I used the same approach to statement 2) (n-3)/7=Q+0 ==>n/7-3/7=Q ==>n/7=Q+3/7, hence remainder is 3.

Is it the right way to solve it?
gmat-admin's picture

Link: https://gmatclub.com/forum/when-a-positive-integer-n-is-divided-by-7-wha...

Your analysis of statement 1 is almost perfect.
The only problem is that we can't get a remainder of 45 when we divide by 7
45/7 leaves a remainder of 3 (which now matches the remainder for statement 2)


Dear Brent,

to find the Remainder of 3^24/5, I just look at the unit digit of the dividend (1) and i can immediately derive that the remainder will always be ONE.

However, in the problem "If N = 775 × 778 × 781, what is the remainder when N is divided by 14?" why cant i find the unit digit of n (ZERO) and derive that the remainder is SIX? This works just for some number that end with 0:
20/14 but not 30/14...

Thank you!
gmat-admin's picture

Your strategy for the first question works because there's a nice relationship between the units digit of integer K and the remainder when K is divide by 5. The same relationship does not exist when we divide by 14.

For example, if K has units digit 6, then K COULD equal 6, 16, 56, 106, 336, 416, etc
For ALL of these possible values of K, we get remainder 1 when we divide by 5

Now let's see what happens when we divide by 14.
Let's take the same POSSIBLE values of K (where K has units digit 6)
6 divided by 14 leaves remainder 6
16 divided by 14 leaves remainder 2
56 divided by 14 leaves remainder 0
106 divided by 14 leaves remainder 8

As you can see, knowing the units digit of K does not help up predict the remainder when we divide by 14.

Does that help?


Therefore, I can use the unit digits method only in case the divisor is 5 (or multiples)?
For example, I used this method to also solve "what is the remainder of 4^(2a+1+b)/10".
gmat-admin's picture

Your approach will work if we're dividing by 2, 5 or 10 (since we have a base 10 number system)



In this question, when we solve the equation, we get y as 7 and when we substitute y as 7 in the equation, x equals to 1, which satisfies all the other options except for C. Could you please help me with the query, thanks!
gmat-admin's picture

Question link: https://gmatclub.com/forum/when-15x-is-divided-by-2y-the-quotient-is-x-2...

Question: When 15x is divided by 2y, the quotient is x, and the remainder is x. If x and y are positive integers, which of the following must be true?

If we solve the resulting equation, the x's disappear, which means x can be ANY positive integer from 1 to 13.
ASIDE: Since x is the remainder, x must be less than 14 (aka 2y).

For example, if x = 10, we get: When 150 is divided by 14, the quotient is 10, and the remainder is 10. Works perfectly!

If x = 11, we get: When 165 is divided by 14, the quotient is 11, and the remainder is 11. Works perfectly!

Does that help?


Hey Brent,

in this Q::


Is it possible to just do 50/3 + 1? Or does that just work because of luck?


gmat-admin's picture

Link: https://gmatclub.com/forum/how-many-integers-from-0-to-50-inclusive-have...

50/3 + 1 = 17.6666...
So, which answer do you choose? (17 or 18)?

That strategy will work some of the time, but not always.

For example, if the numbers are from 1 to 10 inclusive, the correct answer = (10-1)/3 + 1 = 4
That is, there are 4 numbers in that range that leave a remainder of 1 when divided by 3.
Those 4 values are 1, 4, 7, and 10

However, if the numbers are from 2 to 11 inclusive, the correct answer = (10-1)/3 = 3
That is, there are 3 numbers in that range that leave a remainder of 1 when divided by 3.
Those 3 values are 4, 7, and 10



Formula: N ÷ D = Q(R) means Q × D + R = N
I get: n = 18q + 7
Then what?
gmat-admin's picture

Question link: https://gmatclub.com/forum/if-the-remainder-is-7-when-positive-integer-n...

Here's the question: If the remainder is 7 when positive integer n is divided by 18, what is the remainder when n is divided by 6?
A. 0
B. 1
C. 2
D. 3
E. 4

You determined the following: n = 18q + 7 (perfect)
Rewrite 7 as follows: n = 18q + 6 + 1
Factor 6 from the first part: n = 6(3q + 1) + 1

We know that 6(3q + 1) is a multiple of 6
So, 6(3q + 1) + 1 is ONE GREATER THAN a multiple of 6
So, if we divide 6(3q + 1) + 1 by 6, the remainder will be 1

Answer: B


There are two rules

1/If N divided by D equals Q with remainder R, then N = DQ + R If N divided by D equals Q with remainder R, then N = DQ + R"

2/ If N divided by D leaves remainder R,then the possible values of N are R,

my question is *HOW CAN WE CHOOSE Between THESE ONES*?
ISN'T THE ANSWER IS---- if we face questions where Quotient is availabl, we use number one rule?
gmat-admin's picture

In some cases, we can use both properties, and, in other cases, only one property will apply.
Unfortunately, there's no set rule that will tell you which property to use.

If we're told the actual value of the quotient, then it's more likely that the "N = DQ + R" will apply.
If we're not told the actual value of the quotient, then it's more likely that the " possible values" property will apply.

That's the most concrete answer I can provide.
I hope that helps.


Here, you use both rules. Imo,both rules are best.But i'm struggling which i use at first place.
gmat-admin's picture

Question link: https://gmatclub.com/forum/what-is-the-remainder-when-positive-integer-n...

It may take a bit of time to determine which rule is best for any given question.
If you're not sure, just start with one of the approaches and closely monitor your progress. If, after 15 or 20 seconds, you feel that this approach isn't going anywhere, then change to the other approach.


I think here is a typo in E) NOT 41!
gmat-admin's picture

Good catch!
I've edited my response accordingly.

Cheers and thanks for the heads up,


Is there any approach except applying divisibility of 9?
gmat-admin's picture

Question link: https://gmatclub.com/forum/when-m-is-divided-by-9-the-remainder-is-2-whe...

As I mention in my solution, I think Bunuel's approach (https://gmatclub.com/forum/when-m-is-divided-by-9-the-remainder-is-2-whe...) is faster.

That said, my approach (https://gmatclub.com/forum/when-m-is-divided-by-9-the-remainder-is-2-whe...) isn't significantly more time consuming.

Bunuel actually used this
"N = DQ + R"after that he said m can be.....
I Didn't understand that part.i mean how he assume those values? Would you please solve this by that formula?

However, a little bit off-track question. Pardon me!
I am following butler project run by gmat club. In CR butler,they are giving LSAT questions.Tbh,those are huge hard to digest.should i practice them or not?
gmat-admin's picture

Now that I've actually taken a closer read of Bunuel's solution, I prefer my approach :-)

Here's the concept that Bunuel is using:
To set things up, let's say that n divided by 3 leaves remainder 1, and n divided by 5 leaves remainder 2.

Based on the 1st fact, the possible...
...the values of n are: 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37 etc

Based on the 2nd fact, the possible...
...values of n are: 2, 7, 12, 17, 22, 27, 32, 37, etc

7 is the positive smallest n-value that satisfies BOTH conditions.
This corresponds with the r-value that Bunuel noted in his solution.

Now let's look at the n-values that satisfy BOTH conditions.
They are 7, 22, 37, etc
Notice that these values increase by 15 each time.
So, if we were to continue the pattern we would get: 7, 22, 37, 52, 67, 82, etc
We get 15 by finding the product of the divisors for both facts.
That is, 15 = (3)(5)
This corresponds with the p-value that Bunuel noted.

So for the example above, the n-values that satisfy BOTH conditions are in the form: n = 15x + 7, where x is any integer from 0 to infinite.

Regarding LSAT questions...
LSAT critical reasoning questions include both inductive and deductive reasoning.
Inductive reasoning = formal logic (.e.g, "if A then B" implies "if not B then not A")

Very few GMAT CR questions involve inductive reasoning. In fact, I'd say that less than 2% of GMAT CR questions involve inductive reasoning.

Conversely, on the LSAT, there are more inductive reasoning questions than deductive reasoning questions. For this reason, many of the LSAT questions will be quite confusing. If I were you, I'd stick with official GMAT CR questions. There are thousands of such questions on GMAT Club's website.

I hope that helps.

Thanks a ton sir

Find another fraction EQUIVALENT to 435/100
How about 174/40 (= 4.35)
Didn’t get that part.

gmat-admin's picture

Question link: https://gmatclub.com/forum/when-positive-integer-a-is-divided-by-positiv...

As you might guess, there are infinitely-many fractions that are equivalent to 435/100 (aka 4.35).

So, for the above question, the goal is to keep testing equivalent fractions until we get a remainder that matches one of the answer choices.

If we take: 435/100
And divide top and bottom by 5, if we get the equivalent fraction 87/20.
When we divide 87 by 20, we get a remainder of 7
7 is not among the answer choices. HOWEVER, I do see 14 (which is TWICE 7) among the answer choices.
This is a hint that suggests we take 87/20 and multiply top and bottom by 2 to get the equivalent fraction 174/40

At this point, when we divide 174 by 40, we got a remainder of 14, which IS among the answer choices.

Does that help?


******Didn't get this part
Since p < q, then p divided by q equals 0 with remainder p (remainder p how??)
& it will be great helpful if you rephrase this part again.********

Since it's IMPOSSIBLE for the remainder to both EQUAL p and BE LESS THAN p, we can conclude that it's impossible for p to be less than q.

Using similar logic, we can see that it's also impossible for q to be less than p.

So, it MUST be the case that p = q
So, pq = p² = the square of some integer

Check the answer choices . . . only D is the square of an integer.


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