For a start, give these problems a try. A complete explanation will come at the end of the discussion.

**Q1**. When positive integer

(A)

(B)

(C)

(D)

(E)

**Q2**. When positive integer

(A)

(B)

(C)

(D)

(E)

**Q3**.

(A)

(B)

(C)

(D)

(E)

**Q4**. When positive integer

(A)

(B)

(C)

(D)

(E)

I will discuss those questions at the end of this article.

Let’s look at division carefully and think about the parts. Suppose we divide

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 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

That decimal, **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:

We divide the dividend

Virtually any problem on the QUANT that gives you a decimal quotient is relying on this particular formula. It is crucial for answering #1 and #3 above.

Let’s go back to the integer relationships:

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:

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.

**Q1**. We know that

So

So

**Answer = E**

**Q2**. Use the rule

to translate each sentence.

The first sentence becomes

Now that we know **doubling and halving**” trick. Double

**Answer = A**

**Q3**. Here, we have to use

to translate each act of division. The first one tells us

Both are equal to

Now that we know

**Answer = C**

**Q4**.