# The Remainder Of It All

The result of the division of an integer by another can be viewed in many different ways, depending on what the question asks.

For example 5 divided by 3 can be expressed as:

- A Fraction: 5/3
- A decimal: 1.67
- A quotient and an integer remainder: 5 divided by 3 gives a quotient of 1 with a remainder of 2.

The quotient is the whole number result of division i.e. the quotient of 5/3 is 1 and the quotient of 11/2 is 5.

A **remainder** is what’s left after the arithmetic action of **division**.

In the above example, the multiple of 3 that is nearest to, but still smaller than five is 3. The distance between 3 and 5 is the **remainder** of 2.

Note that when 8 is divided by 3, the remainder – the distance between 8 and the nearest multiple of 3 (which is 6) – is still 2.

**Remainder** is defined as the distance (in units) from the dividend to the nearest multiple of the divisor that is smaller than the dividend.

Let’s try a simple example:

Which of the following is the remainder when dividing 13 by 5?

A) 1

B) 2

C) 3

The nearest multiple of 5 that is smaller than 13 is 5 times 2 = 10. The distance between 10 and 13 is 3.

Note that since the remainder is measured from the nearest multiple of the divisor, the greatest **possible remainder** is one less than the divisor. The **remainder** can never be equal to, or greater than the divisor. In the above example, a remainder of “8″ when dividing by 5 is illogical, because 8 is greater than 5, and includes another multiple of 5 closer to the dividend.

Likewise, dividing 10 by 5 does not give a remainder of 5 because 10 itself is the nearest multiple of 5 that is closest to itself, giving a remainder of zero (i.e. no remainder).

Identifying **remainder** problems in the GMAT is easy – the question uses the word **remainder** (Duh!). GMAT problems involving **remainders** can usually be solved easily by **Plugging in** numbers that fit the problem.

Let’s try another example:

When x is divided by 5, the remainder is 3. When y is divided by 5, the remainder is 4. What is the remainder when x + y is divided by 5?

A) 0

B) 1

C) 2

D) 5

E) 7

Plug in numbers for x and y that fit the problem: x = 8 and y = 9. These numbers give a remainder of 3 and 4, respectively when divided by 5.

x+y = 8+9 = 17. When 17 is divided by 5, the closest multiple of 5 is 15, and the remainder is 2.

You can plug in a different set if you’re not sure. Try *x* = 13 and *y* = 14 – the remainder of *x*+*y* will still be 2 when divided by 5.

Note that you could have easily just plugged in the remainders themselves: *x* = 3 and *y* = 4 would’ve also worked. The nearest multiple of 5 that is smaller than 3 is zero; the distance between zero and 3, is 3, so the remainder when dividing 3 by 5 is indeed 3. The same goes for 4: the remainder when dividing 4 by 5 is 4 (the difference of 4 from zero). Thus, *x* = 3 and *y *= 4 are also valid plug ins. How’s that for an easy calculation?

Where plugging in is difficult to use because the question requires large numbers, use the following equation:

For any integer *i* divided by another integer *d*

*i* = quotient·*d* + remainder

Thus, when *i* is divided by 5 the remainder is 3, *i* can be expressed as the equation:

*i *= 5*x *+ 3

Here *x* is the quotient.

This form also supports plugging in: plug in values for *x*, and you will find the corresponding values of *i*:

If *x *= 0 –> *i *= 5⋅0 + 3 = 3

If *x *= 1 –> *i *= 5⋅1 + 3 = 8

If *x *= 2 –> *i *= 5⋅2 + 3 =13

and so on.

**Remember:**

Remainder is the distance (in units) from the dividend to the nearest multiple of the divisor that is smaller than the dividend.

Now try solving this question:

If

i,aandbare integers, is 4(3b+ 2) = 5a?(1) If

iis divided by 5 the quotient isaand the remainder is 3(2) If

iis divided by 12 the quotient isband the remainder is 11

## 4 comments

Luke Lee on October 8th, 2012 at 9:45 pm

"Plug in numbers for x and y that fit the problem: x = 7 and y = 9. These numbers give a remainder of 3 and 4, respectively when divided by 5." There is one small error here, x = 8, in order to get a remainder of 3.

As for the question at the end, the answer should be C.

From (1) i = 5a + 3, which gives us the RHS of the equation as i - 3. From (2) i = 12b + 11, which gives us the LHS of the equation as i - 11 + 8, or i - 3. LHS = RHS, so 4(3b+2) = 5a.

Faruk on October 11th, 2012 at 6:51 am

First,organize the question

4(3b + 2) = 5a

we get 5a - 12b =8

st 1 gives i = 5a + 3

st 2 gives i - 12b + 11

so 5a + 3 = 12b +11 ,we get 5a -12b = 8 answer is C

Denysse on December 9th, 2012 at 11:42 am

I don't understand how statement 1 and 2 gives you those equations. I would think statement one gives you i/5= a + 3 and then you multiply the entire thing by 5 so it would become i = 5a + 15 and then similar to statement 2?

Can someone please help?

Evan on June 28th, 2013 at 3:38 am

Better to apply the formula:

a/b = c + r/b

So:

i/5 = a + 3/5

i/12 = b + 11/12

We can then get both a and b in terms of i, plug back into the original question, solve for i, then solve for a and b. Of course, we don't need to perform the calculations. We just need to determine that we can, which is the case when we have both the information in (i) and (ii). So (C).