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.

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)


hi Brent, i am confused about the fact if the dividend is lower than the divisor then, the remainder always equals the dividend. so 8/11 = 0 with remainder 8 where the answer is actually 0.72. where did the 8 come from?
same with 3/4. is the remainder 3 in this case? even though 3/4 = 75/100 = 0/75
gmat-admin's picture

Hi tsimo,

I think you are confusing two different concepts (fractional notation and decimal notation).

When it comes to dividends, divisors, remainders and quotients, we're looking at how many times a value (the divisor) will divide into the dividend.

So, for example, 5 divides into 17 THREE times and we're left with a remainder of 2.
In fractional terms, we can write: 17/5 = 3 2/5

Likewise, 7 divides into 36 FIVE times and we're left with a remainder of 1.
In fractional terms, we can write: 36/7 = 5 1/7

Likewise, 11 divides into 6 ZERO times and we're left with a remainder of 6.
In fractional terms, we can write: 6/11 = 0 6/11

On the other hand, converting a fraction to a decimal basically involves finding an equivalent fraction where the denominator is a power of 10.

For example, 1/2 = 5/10 = 0.5
And 3/4 = 75/100 = 0.75
And 7/25 = 28/100 = 0.28
And 5/8 = 625/1000 = 0.625

Does that help?


got it! makes perfect sense. thank you!

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


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