For a start, give these problems a try. A complete explanation will come at the end of the discussion.
1) When positive integer N is divided by positive integer J, the remainder is 14. If N/J = 134.08, what is value of J?

(A) 22
(B) 56
(C) 78
(D) 112
(E) 175
2) When positive integer N is divided by positive integer P, the quotient is 18, with a remainder of 7. When N is divided by (P + 2), the quotient is 15 and the remainder is 1. What is the value of N?

(A) 151
(B) 331
(C) 511
(D) 691
(E) 871
3) P and Q are both positive integers. When P is divided by Q, the remainder is some positive integer D, and when P is divided by (Q + 3), the remainder is also D. If P/Q = 1020.75 and P/(Q + 3) = 816.6, then which of the following gives the correct set of {D, Q}?

(A) {6, 12}
(B) {6, 15}
(C) {9, 12}
(D) {9, 15}
(E) {15, 24}
I will discuss those questions at the end of this article.
Division Terminology
Let’s look at division carefully and think about the parts. Suppose we divide 33 by 4. Of course, 4 goes into 33 eight times, with a remainder of 1. Let’s talk about the official names of this cast of characters.
33 = the dividend = the number being divided
4 = the divisor = the number doing the dividing; the number by which you divide
8 = the integer quotient = the integer that results from a whole number of divisions
1 = the remainder
We have to add a caveat here. Notice: here we are talking about positive integers only — living in the magical fairyland where the only numbers that exist are positive integers, where skies are not cloudy all day. Unless you live on a farm where the barnyard animals all sing in unison, you don’t get to stay here forever.
Of course, numbers in the real world aren’t like that, and if you prance through the GMAT Quantitative Section as if it’s a magical fairyland where all the numbers are positive integers, this section will utterly decimate you. In the real world that involves all possible numbers, this process looks a bit different. For example, if you type 33/4 into your calculator, your calculator will tell you
33/4 = 8.25
That decimal, 8.25, is the quotient, the actual realworld mathematical quotient. In this article, for clarity, I will refer to this one as the “decimal quotient.”
Notice, first of all — the integer part of the decimal quotient is exactly equal to the integer quotient. It has to be. In fact, we can go a little further: Let’s look at this process both with words and with numbers:
\({dividend/divisor} = decimal quotient = integer quotient + {remainder/divisor}\)
\({33/4}= 8.25 = 8 + {1/4}\)
We divide the dividend (33) by the divisor (4), and we get the decimal quotient. The integer part of the decimal question is the first piece of the last sum, the integer quotient. What’s crucially important is — the decimal part of the decimal quotient equals the final fraction:
\(decimal part of decimal quotient ={remainder/divisor}\)
Virtually any problem on the GMAT that gives you a decimal quotient is relying on this particular formula. It is crucial for answering #1 and #3 above.
Rebuilding the Dividend
Let’s go back to the integer relationships:
\({dividend/divisor} = integer quotient + {remainder/divisor}\)
If you are given the divisor, the integer quotient, and the remainder, then you can rebuild the dividend. In particular, notice that “divisor” is the denominator of both fractions, so we if multiple all three terms by “divisor”, it cancels in two of the three terms:
dividend = (integer quotient)*(divisor) + remainder
That formula is pure gold in questions which give you an integer quotient, a divisor, and a remainder. Even if one or two of those three are in variable form, it allows us to set up an algebraic relationship we can solve. This is crucial for answering #2 above.
Practice Questions
It may be, at this point, you want to give those three questions another attempt before reading the solutions. Here’s yet another practice question:
4) http://gmat.magoosh.com/questions/305
Practice Question Solutions
1) We know that
\(decimal part of decimal quotient ={remainder/divisor}\)
So 0.08, the decimal part of the decimal quotient, must equal the remainder, 14, divided by the divisor J.
0.08 = 14/J
0.08*J = 14
J = 14/0.08 = 1400/8 = 700/4 = 350/2 = 175
So J = 175, answer = E.
2) Use the rule dividend = (integer quotient)*(divisor) + remainder to translate each sentence.
The first sentence becomes N = 18P + 7. The second equation becomes N = (P + 2)*15 + 1, which simplifies to N = 15P + 31. These are ordinary simultaneous equations (https://magoosh.com/gmat/math/algebra/gmatquanthowtosolvetwoequationswithtwovariables/). Since they both equal N already, let’s set them equal and solve for P.
18P + 7 = 15P + 31
3P + 7 = 31
3P = 24
P = 8
Now that we know P = 8, we can just plug in. The product 15*8 is particularly easy to do, without a calculator, by using the “doubling and halving” trick. Double 15 to get 30, and take half of 8 to get 4 — 15*8 = 30*4 = 120. So
N = 15(8) + 31 = 120 + 31 = 151
N = 151, answer = A.
3) Here, we have to use
\(decimal part of decimal quotient ={remainder/divisor}\)
to translate each act of division. The first one tells us 0.75 = D/Q, and the second one tells us that 0.60 = D/(Q + 3). These are also two simultaneous equations. Multiply both to get rid of the fractions.
D = 0.75*Q
D = 0.60*(Q + 3) = 0.60*Q + 1.80
Both are equal to D, so set them equal to each other and solve for Q.
0.75*Q = 0.60*Q + 1.80
0.15*Q = 1.80
Q = 1.80/0.15 = 180/15 = 60/5 = 12
Now that we know Q = 12, we can find D = 0.75*Q = (3/4)*12 = 9. So D = 9 and Q = 12. Answer = C.
What if the question has three unknowns? For example,
When n is divided by 14, the remainder is 10. What is the remainder when n is divided by 7? In this problem, the equation cannot be directly used.
A problem like this would be unsolvable without more information, because there is more than one number that can be divided by 14 to get a remainder of 10. In this case, 24/14 is 0r1 and 4/14 is 1r10. But 24/7 is 0r17 and 4/7 is 1r3.
You could, on the other hand, have a question like this that read “What is a possible remainder when n is divided by 7.” Because it is possible to narrow down n to a finite number of possible values.
For that matter, if n were being divided by 7 to begin with, with a given remainder of 10, the only possible value for n would be 17.
In short a question with three unknowns could be solvable down to a set of possible answers, or solvable to one answer, or virtually impossible to solve, depending on the numbers that were used. 🙂
Hello,
I just took the GMAT a few days ago and came across a remainder question that appeared fairly simple. The question was something as follows, “what is the sum of all the remainders of positive integers less than or equal to 48 divided by 6”
I did the following and quickly realized there was a pattern.
6/6=1 r 0
7/6=1 r 1
8/6=1 r 2
9/6=1 r 3
10/6=1 r 4
11/6=1 r 5
12/6=2 r 0
13/6=2 r 1
etc…
Obviously this pattern repeats all the way to 48 so it is easy to simply add up the remainders. I thought the answer would be 105. However, then I started thinking that numbers like 0,1,2,3,4 and 5 divided by 6 also have remainders. So essentially my question is as follows…
does 1/6 have remainder 5, 2/6 have remainder 4, 3/6 have remainder 3, etc.
Hi Hunter,
You cannot have a remainder larger than the number you started with. The remainder of 1/6 is just 1. The remainder of 2/6 is 2. And so on. I hope this clarifies! 🙂
Hello,
So, is 105 the right answer?
i think answer will be 120. when every number less or equal of 48 is divided by 6 remainders will follow a pattern as 1, 2, 3, 4, 5, 0 and upto 48 this pattern will repeat 8 times( 48/6=8). again some of 1, 2, 3, 4, 5, and 0 =15. so some of all remainder upto 48 will be 15*8= 120. i hope that is correct..
You’ve got it, Julkar. Correct answer, perfect logic. 🙂
Dear Mike,
you are an amazing teacher.I wish my grade school teachers were like you
Mr. Mike Mcgarry,
I feel myself as fortuitous as i got the opportunity to learn something from your lessons.
your MULTIPLIER is one of the greatest groundbreaking shortcuts that i learned here and now today the remainder/divisor !! Just simply outstanding. I never get bored in your writing because it’s always been a brainstorming to do the lessons and i myself think that, at least in MATH, failure to solution is necessary to excel in. My words are verbose but not enough to show my gratitude, SIR !
Dear Pranesh,
Thank you very much for your kind words! 🙂 It means a great deal to me to know that you have learned from my blogs! 🙂 My friend, I wish you the greatest of good fortune in all your studies!
Mike 🙂
2) I multiplied a natural number by 18 and another by 21 and added the products.which one of the followig could be the sum?
a)2007 b)2008 c)2006 d)2002
a. 2007
sir ,I could not solve these problems
1) 7 is added to a certain number,the sum is multiplied by 5,the product is divided by 9 and 3 is substracted from the quotient.Thus if the remainder left is 12,What is the original number?
20
This is beautiful. I’ve never seen such before and am very grateful.
I have a question about # 2. I just do not see where you got the 31 from. I bet if you explain I will have that ah ha moment. please help
Why did you put the 15 to the right side of P+2 instead of leaving it at the left side?
I want to know so that I can apply that to future problems.
Dear Elie,
I’m happy to respond. 🙂
First of all, understand that we can change the order of multiplication, and this doesn’t change the results (that is called the commutative property of multiplication) — 4*6 = 6*4 = 24. Similarly, 15*(P + 2) = (P + 2)*15 — the order doesn’t matter at all, so those two are entirely the same an interchangeable.
Now, that equation I had in the solutions of #2 was: N = (P + 2)*15 + 1. If we distribute the 15 through the parentheses, we get: N = 15P + 30 + 1. When we add the 30 and the 1, we get the 31.
Does all this make sense?
Mike 🙂
I think it would be very helpful but my query Is…is it enough for GRE ?Thank you 🙂
Dear Nur,
I’m happy to respond. 🙂 If you understand everything in this blog, you will be more than prepared for anything that would appear on the GRE Quant section on remainders.
Does this answer your question?
Mike 🙂
Hi, if I fully understand simultaneous equations, would it be sufficient to solve these type of questions?
Dear Kailee,
If you know the ideas on this page (decimal – remainder relationship, rebuilding the dividend, etc.), then the GMAT will throw nothing harder at you than can be handled by straightforward simultaneous equations. Does this make sense?
Mike 🙂
Dear Mike
I have a question about (1).
If the decimal part of the decimal quotient is remainder/divisor, then why is 0.08 (2/25) not R(remainder)/D ?
Divisor J must be equal 25.
but answer is J =175.
Dear Motoki,
I’m happy to respond. 🙂 This is always a tricky thing about ratios — if we know the ratio of A/B, we don’t necessarily know the individual values A & B. For example, if we know that the ratio of males to females at a company is 3:2, that doesn’t mean that just five people work at that company! 🙂
Much in the same way, in this problem, it absolutely must be true that remainder/divisor is in the ratio of 0.08 = 2/25, but it would be a HUGE mistake to deduce from this fact that the remainder must be 2 and the divisor must be 25. In fact, we are told that the remainder is 14. All we know from that decimal part of the division is a ratio, not individual quantities.
This mistake about ratios is a very easy mistake to make, a very tempting trap in a variety of GMAT situations involving ratios. Always be very discerning about ratios — about exactly what you do know and exactly what you don’t know. When you understand how to think about ratios and execute proportional thinking on the GMAT, that’s an enormously powerful problemsolving tool.
Does all this make sense?
Mike 🙂
Thank you Mike.
Finally I could comprehend the ratio!!!
Dear Motoki,
You are quite welcome. 🙂 Best of luck to you!
Mike 🙂
Thanks Mike you’re phenomenal as always. This is a great lesson
But I’m intrigued– why have we not seen this rule before
Meaning, decimal part = remainder/divisor… i haven’t seen it anywhere
Dear Aisha,
Thank you for your kind words. I’m very glad you have found this helpful. Yes, it’s funny: not many people talk about this math factoid. It doesn’t often show up on the GMAT, and when it does, it is usually on the harder math questions. It’s peculiar that no one else mentions it, because it could be important on a question.
Mike 🙂
Dear Mike,
Thanks for all your great methods and suggestions. It was of great help to me and it made me nail my Gmat. Thanks!
Niels,
Congratulations! You are more than welcome! Best of luck in your future!
Mike 🙂
I am confused about a part of the explanation given for problem three. If the decimal part of the decimal quotient is remainder/divisor, then why is 0.75 D/Q and not R(remainder)/D ?
Nadeen,
That problem is a little confusing because it uses different letters. Notice, in that problem D is the remainder and Q is the divisor. Don’t be fooled by letters alone — you always have to look at the concepts behind the letters.
Mike 🙂
very nice, very helpful
Sadie,
Thank you for the kind words. Best of luck to you.
Mike 🙂
Dear Mike, Thanks for yet another terrific blog post!
BTW, how does this shortcut to solve Q # 3 look? 🙂
Since, it is given P/Q = 1020.75 and remainder is D, D/Q= 0.75 which is 3:4. The only answer choice that satisfies this condition is ‘C’
Thanks!
Dear Mensa Member,
Yes, that’s a great shortcut. Of course, if this had been a true GMATstyle question, probably there would have been more than one choice for which that ratio was true. I didn’t think to include that when I wrote the question, but that’s precisely the kind of thing they would see.
Does this make sense?
Mike 🙂
Yes, that would be a deadly GMAT trap, if one was not careful. Thanks, Mike, for the insight!
You are more than welcome.
Best of luck to you.
Mike 🙂
That’s what I did as well, and I would have chosen “C” on the GMAT, but I doubted because it seems to completely disregard the second remainder/divisor which is 60/100 or 3/5. How does that come into play?
Dear Katie,
I’m happy to respond. 🙂 Remember, in that second division, the divisor is (Q + 3). We would have to add three to the second number in each answer pair, and then check the ratios — for {9/12}, the division we would check is 9/(12 + 3) = 9/15 = 3/5.
Does this make sense?
Mike 🙂
Hi Mike,
I did not understand the below part. In words, it says divisor/dividend but in numbers dividend/divisor which is actually correct. Could you please elaborate more on this as in connection between word and number formula below?
{divisor/dividend} = decimal quotient = interger quotient + {divisor/dividend}
{33/4}= 8.25 = 8 + {1/4}
Dear Sarika,
I’m sorry. There were a couple typos in that line, I believe I have corrected everything. Thank you for pointing this out, and let me know if there is anything that still doesn’t make sense.
Mike 🙂
Thank you. I studied four semesters of math while in college (25= years ago) and never learned this. Frightening. But wow will this help with problems like this on the test.
Jeff — it’s funny — I think most of us heard about this when we were in late grade school, but at that point, mathematical niceties were lost on us, and then going through puberty blurred everything. Unfortunately, many math teacher do not return to these basic subjects when students are older and able to understand the underlying logic. For example, it helps to think about this topic with algebraic symbolism, but when we first learned it in the fourth grade, we hadn’t learned algebra yet! I’m glad you found this helpful. Thank you for your kind words.
Mike 🙂
Thanks! Thanks to you I finally got a good method for remainders problem.
You are quite welcome.
Mike 🙂
Very nice set of material here. Thanks for the information.
Thank you for your kind words. Best of luck to you.
Mike 🙂