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Remainder when dividing negative numbers

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by MBA.Aspirant » Thu Jun 16, 2011 11:29 am
I want to understand how the remainder works when you divide a negative integer by another negative one or by a positive one.

For example, -11/6 is it equal -2 remainder 1 or how does it work exactly?

I take it that the remainder is either > or = 0, but some guys are saying it can be negative. Checkout this thread: https://gmatclub.com/forum/finding-the-r ... 88839.html

Thanks
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by winniethepooh » Thu Jun 16, 2011 12:23 pm
A remainder can be negative.
Like in Your example -11/6 will give -5 as the remainder and the quotient will be -1.
See it this way -11 = 6 x(-1) + (-5).
When you divide -11/-6 the remainder will be -5 and the quotient will be 1(positive
See it this way -11 =(-6) x1 + (-5)

Hope this helps!
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by winniethepooh » Thu Jun 16, 2011 12:25 pm
Well, this is possible only with integers and not real numbers. In real numbers there's no remainder!
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by MBA.Aspirant » Thu Jun 16, 2011 12:44 pm
I have the ETS math review open now and it says " The remainder is always greater than or equal to 0 and less than b" where b is the divisor.

Also why can't it be -2 remainder 1? -11 = (-2*6) +1= -11
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by Brent@GMATPrepNow » Thu Jun 16, 2011 12:45 pm
Don't worry, the GMAT will not ask remainder questions involving negative integers.
They always include a condition that the numbers are positive integers (unless this is implied in the question)

From the OG12: If x and y are positive integers, there exist unique integers q and r, called the quotient and remainder, respectively, such that y = xq + r and 0 < r < x.

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by Brent@GMATPrepNow » Thu Jun 16, 2011 12:47 pm
Aside: If you're converting a negative entire (improper) fraction to a mixed fraction, just ignore the negative sign and add it in later.

Example, -17/5 = -3 2/5

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by winniethepooh » Thu Jun 16, 2011 1:15 pm
That is true, but to answer your question whether remainder can be negative, the answer is YES.
Even as per Brent's explanation -17/5 = -3 2/5 = converting again into fraction will give -3 x 5 + (-2) = -17/5.
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by MBA.Aspirant » Thu Jun 16, 2011 1:35 pm
So to calculate the remainder of -11/6, do you let the quotient be -2 and r 1, or quotient -1 and r -5?
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by Ian Stewart » Thu Jun 16, 2011 1:45 pm
Remainders, by definition, are *never* negative. If you are dividing by 6, for example, there are only six possible remainders: 0, 1, 2, 3, 4 or 5.

If you want to understand what the remainder would be when you divide, say, -7 by 6, we can first see how we find remainders with positive numbers. When you divide, say, 19 by 6, the remainder is 1 because 19 is 1 greater than the nearest smaller multiple of 6. That is, we need to subtract 1 from 19 to get a multiple of 6. Similarly, if we try to divide -7 by 6, the remainder will be 5, because -7 is 5 greater than the nearest smaller multiple of 6 (which is -12). That is, we need to subtract 5 from -7 to get a multiple of 6.

As Brent points out, you will only ever be asked to find remainders when dealing with positive numbers on the GMAT. On very rare occasions, if you want to apply 'modular arithmetic' (if you don't know what that is, you don't need to know - it's one of many possible approaches to remainders questions), you might want to know what remainders you get when you divide negative numbers by positive numbers, but it's certainly only important on very rare questions, if at all.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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by winniethepooh » Thu Jun 16, 2011 1:46 pm
Yes, both stand correct.
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by winniethepooh » Thu Jun 16, 2011 1:54 pm
https://en.wikipedia.org/wiki/Remainder# ... l_integers

This says what I want to prove.
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by MBA.Aspirant » Thu Jun 16, 2011 2:52 pm
Thanks to all!

This came up in a GRE question:

The remainder when n is divided by 3 is 1, and the remainder when n+1 is divided by 2 is 1.

Quantity A) The remainder when n-1 is divided by 6
Quantity B) 3

I) A is bigger
II) B is bigger
III) Both are equal
IV) Insufficient info


If you try n =10 you get that the remainder is 3. But if you try n= -10 the remainder is 1 with quotient -2 or -5 with quotient -1
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by MBA.Aspirant » Thu Jun 16, 2011 3:03 pm
Question:

-10 /3

remainder = 2 with quotient -4, or -4 with quotient -2, or -7 with quotient -1?
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by Brent@GMATPrepNow » Thu Jun 16, 2011 3:12 pm
MBA.Aspirant wrote:Question:

-10 /3

remainder = 2 with quotient -4, or -4 with quotient -2, or -7 with quotient -1?
If we're talking about remainders as far as the GRE is concerned, then it is possible to have (GRE) questions involving negative dividends and divisors. But, as Ian said, the remainder must be either zero or positive.

So, -10 divided 3 is equal to -4 with remainder 2

But, to be clear, the GMAT will not require us to consider negative dividends and divisors.

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