Remainders are the NUMERATOR of a fraction from a mixed number that results from division. For example, 19/3 leaves a remainder of 1, since 19/3  = 6 1/3.

Some quick tips:

1. Your remainder can only range from zero to the denominator of the fraction. For example, when dividing by 9, your remainder options are 0-8, since a remainder of 9 leaves you a new whole number (with a remainder of 0).
2. Look for the closest whole number and count up or down from there. For example, when trying to find the remainder of 146/15, you can see that 15 would go into 150 evenly. You then count down four from 150 to 146, so your remainder is (15 -4) = 11. This is easier than recognizing 135/15 is a whole number and counting up.
3. Become familiar with common trends or patterns. For example, multiples of even numbers are even, so 167/(even #) must have an ODD remainder.
4. The remainder should NOT be reduced. 18/4 = 4 2/4. The remainder stays equal to 2, even though you can reduce 4 2/4 to  4 1/2.

I recently came across this question, which I think is a good introduction:

What is the remainder of 3^(4n+3) divided by 5, assuming n is a positive integer?

Firstly, when dividing by 5, we are looking for the remainder above a one’s digit of either 0 or 5. In this scenario, we only care about the one’s digit, so we only need to look at the one’s digit while multiplying.

We can break 3^(4n+3) into 3^4n * 3^3 by the rules of exponents.

If n = 1, 3^4n = 3^4 = 81 = one’s digit of 1.

If n = 2, 3^4n = 3^8 = 81*81 = one’s digit of 1.

We detect the pattern that regardless the value of n, we will be multiplying a term with a one’s digit of 1 with a term with a one’s digit of 7 (3³), so the result will have a one’s digit of 7. When and number with a one’s digit of 7 dividing by 5, we are left with a remainder of 2.

Pattern questions with division are many times Remainder questions at their core

The 4 members of the Jones Family rotate who takes out the trash on a daily basis. The order goes as follows: Mom, Dad, Brother, Sister. If Dad takes out the trash on January 18th, who takes out the trash on March 26th? (There are 31 days in January and 28 days in February.)

It’s clear that we don’t want to whip out our calendars and start counting. (A general rule of thumb is that if you think it’s taking too long, it probably is….)

Instead, we see how many days pass between January 18 and March 26:

January 19-31: 13 +

February 1-28: 28 +

March 1 – 26:  26  = 67 days.

67/4 leaves you will a remainder of 3, so we count 3 from Dad, leaving us with Mom on March 26th.

Fractions and Decimals are the same thing

You should be familiar with common decimals, mainly:

1/2 = .5

1/3 = .33 repeating

1/4 = .25

1/5 = .20

1/6 = .166 repeating

1/8 = .125

1/9 = .11 repeating

Note that multiplying these by constants will leave similarly instructive results, such as:

3/8 = 3*1/8 = 3*0.125 = 0.375

The more familiar with these you become, the quicker you can eliminate answer choices are clearly wrong. For example:

If x is an integer, which of the following is a possible value of (x² +2x – 7)/9?

A. 0.268

B. 4.555 repeating

C. -2.4

D. 1.166 repeating

E. 8.125

We don’t have to start plugging in. We know that when divided by 9, the remainder will be a number repeating to the right of the decimal place. Only choice (B) fits that description. ((C) is divided by a factor of 5, (D) by a factor of 6, and (E) by a factor of 8.)

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