In this lesson, we are going to learn how to find the **Greatest Common Factor (GCF)** of the algebraic expressions: **16s³t**, **40s⁵**, and **68t²**. The GCF is the largest factor that divides all given terms without leaving a remainder.

Let’s go through the solution step by step, using a simple and straightforward approach.

### Step 1: Break down each term into its prime factors

Start by factoring each coefficient (the numbers in front of the variables) and the variables themselves.

#### a. 16s³t:

**Coefficient (16)**: The prime factorization of 16 is: 16=2^{4}

**Variables**: For **s³t**, we have: s^{3}=s×s×s,t = t

*So, 16s³t = 2 ^{4}×s^{3}×t*

#### b. 40s⁵:

**Coefficient (40)**: The prime factorization of 40 is: 40=2^{3}×5

**Variables**: For **s⁵**, we have: s5=s×s×s×s×s

* So, 40s⁵ = 2 ^{3}×5×s^{5}*

#### c. 68t²:

**Coefficient (68)**: The prime factorization of 68 is: 68=2^{2}×17

**Variables**: For **t²**, we have: t2=t×t

* So, 68t² = 2 ^{2}×17×t^{2}*

### Step 2: Identify the common factors

Now, we need to find the common factors from each term.

**Coefficients**: Let’s look at the numbers first.

- In 16, the factor is 2
^{4}

- In 40, the factor is 2
^{3}×5.

- In 68, the factor is 2
^{2}×17

^{2}(since 2 is the only factor common to all three, and the smallest power of 2 is 2

^{2}).

**Variables**:

**s-term**: The powers of **s** in the expressions are:

- 16s³t: s3s³s3
- 40s⁵: s5s⁵s5
- 68t²: No
**s**here.

**s**in 68t²,

**s**is not a common factor.

**t-term**: The powers of **t** are:

- 16s³t: t1t¹t1
- 40s⁵: No
**t**here. - 68t²: t2t²t2

**t**common between 16s³t and 68t² is t1t¹t1, so

**t¹**is a common factor.

### Step 3: Multiply the common factors

Now that we have identified the common factors, we can multiply them to find the GCF.

- Coefficient common factor: 2
^{2}=4 - Variable common factor: t
^{1}=t

So, the **GCF** of 16s³t, 40s⁵, and 68t² is: *4t*

### Final Answer: GCF = 4t

### Recap of the steps:

**Factor the coefficients**of each term into primes.**Identify the lowest powers**of the common factors for both the numbers and the variables.**Multiply the common factors**to find the GCF.

Thus, the GCF of **16s³t**, **40s⁵**, and **68t²** is **4t**. This means 4t is the largest factor that divides all three expressions without a remainder.

### Practice Problem

Now, try finding the GCF of the following expressions:

- 18a²b, 24a⁴, and 30ab².

Use the same steps to break down each term and find the greatest common factor. Good luck and

If you feel difficulty finding the GCF then ask in the comment section or use our Online Free GCF Calculator