**High School Number Sense Lesson 37: Finding the Remainder of an Expression**

Hands down, this is one of the most-often concepts tested in number sense, and one of the greatest. If you understand the steps, you can save lots of time coming up with a solution. Often, what would be impossible to solve mentally becomes quite doable. RemainExp showed up

**30 times**on high school tests last year, with a median appearance at

**question # 39**.

**Number Dojo Level: 207**

To begin solving these problems, I need you to trust me on something:

**the remainder after operations = the operations of the remainders**. What I mean is: if you take the remainder of each term of the expression, and then perform the same operations on those remainders as you see in the original expression, you will have the same remainder as if you solved the expression first and then took the remainder.

No matter how I explain this, it will not become clear until we see a few examples.

**Example 1: (59 + 13 - 16) ÷ 5 has a remainder of ___**

- Take the remainder of each term in the expression. 59 ÷ 5 has R
**4**. 13 ÷ 5 has R**3**. 16 ÷ 5 has R**1**. - Plug the remainders back into the expression to get
**(4 + 3 - 1)**, which equals**6**. - Take the remainder of this result:
**6 ÷ 5**has a remainder of**1**, which is your answer.

__: If you solve the expression first, you get (59 + 13 - 16) = 56. 56 ÷ 5 has R__

**Note****1**.

*Don't do this on a number sense; I'm simply illustrating that this method is accurate*.

**Example 2: (41 + 9 x 12) ÷ 7 has a remainder of ___**

- Find the remainder of each term in the expression. 41 ÷ 7 has R
**6**. 9 ÷ 7 has R**2**. 12 ÷ 7 has R**5**. - Plug the remainders back into the expression to get
**(6 + 2 x 5)**, which is (6 + 10) or**16**. - Take the remainder of this result:
**16 ÷ 7**has a remainder of**2**, which is your answer.

__: If you solve the expression first, you get (41 + 9 x 12) = (41 + 108) = 149. 149 ÷ 7 has R__

**Note****2**.

**Example 3: (23 + 45 x 67) ÷ 8 has a remainder of ___**

- Find the remainder of each term in the expression. 23 ÷ 8 has R
**7**. 45 ÷ 8 has R**5**. 67 ÷ 8 has R**3**. - Plug the remainders back into the expression to get
**(7 + 5 x 3)**, which is (7 + 15) or**22**. - Take the remainder of this result:
**22 ÷ 8**has a remainder of**6**, which is your answer.

__: If you solve the expression first, you get (23 + 45 x 67) = (23 + 3015) = 3038. 3038 ÷ 8 has R__

**Note****6**.

**Example 4: 8! ÷ 9 has a remainder of ___**

- This is actually more of a logic problem. 8! is the same as 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1.
- Notice the factors of 6 and 3. 6 x 3 = 18, and 18 ÷ 9 has R
**0**. - No matter what the other factors (remainders) are, if you multiply them by 0, you'll still get 0. So the answer is
**0**.

**Example 5: a/6 has a remainder of 4; b/6 has a remainder of 5. So ab/6 has a remainder of ___**

- Plug the remainders back into the expression to get
**(4)(5)**, which is**20**. - Take the remainder of this result:
**20 ÷ 6**has a remainder of**2**, which is your answer.

**Now let's step this up a notch and include exponents.****Example 6:**

**8**

^{5}÷ 6 has a remainder of ___1. Find the remainder of each term in the expression. 8 ÷ 6 has R

**2**.

2. Plug the remainder back into the expression to get

**2**, which is

^{5}**32**.

3. Take the remainder of this result:

**32 ÷ 6**has a remainder of

**2**, which is your answer.

**Example 7:**

**(6**

^{4}+ 9^{4}) ÷ 5 has a remainder of ___1. Find the remainder of each term in the expression. 6 ÷ 5 has R

**1**. 9 ÷ 5 has R

**4**.

2. Plug the remainders back into the expression to get

**(1**, which is

^{4}+ 4^{4})**(1 + 256)**, which is

**257**.

3. Take the remainder of this result:

**257 ÷ 5**has a remainder of

**2**, which is your answer.

**Example 8:**

**9**

^{10}÷ 11 has a remainder of ___1. Find the remainder of each term in the expression. 9 ÷ 11 has a remainder of

**-2**.

2. Plug the remainder back into the expression to get

**(-2)**, which is equal to

^{10}**2**because of the even power 10.

^{10}3. You may have memorized the fact that

**2**, but in case you don't, break 2

^{10}= 1024^{10}into

**2**, which is

^{5}x 2^{5}**(32 x 32)**.

4. Find the remainder of each term in the expression. 32 ÷ 11 has R

**10**.

5. Plug the remainders back into the expression to get

**(10 x 10)**, which is

**100**.

6. Take the remainder of this result:

**100 ÷ 11**has a remainder of

**1**, which is your answer.

**Here's a free worksheet to help you practice RemainExp:**

remainexp.pdf |

**Up Next for High School: MultFracWhol2**