Prime Factorization Worksheets - Math Monks - Free Printable
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Step-by-step solution for: Prime Factorization Worksheets - Math Monks
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Show Answer Key & Explanations
Step-by-step solution for: Prime Factorization Worksheets - Math Monks
To solve the problem of finding the prime factors of each number and writing them in exponential form, we will break down each number into its prime factors step by step. Let's go through each number:
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- Start with the smallest prime number, 2:
- \( 20 \div 2 = 10 \)
- \( 10 \div 2 = 5 \)
- 5 is a prime number.
- Prime factorization: \( 20 = 2^2 \cdot 5^1 \)
- Exponential form: \( 2^2 \cdot 5 \)
Answer: \( 2^2 \cdot 5 \)
---
- Start with 2:
- \( 72 \div 2 = 36 \)
- \( 36 \div 2 = 18 \)
- \( 18 \div 2 = 9 \)
- Now, 9 is not divisible by 2, so move to the next prime number, 3:
- \( 9 \div 3 = 3 \)
- \( 3 \div 3 = 1 \)
- Prime factorization: \( 72 = 2^3 \cdot 3^2 \)
- Exponential form: \( 2^3 \cdot 3^2 \)
Answer: \( 2^3 \cdot 3^2 \)
---
- Start with 2:
- \( 68 \div 2 = 34 \)
- \( 34 \div 2 = 17 \)
- 17 is a prime number.
- Prime factorization: \( 68 = 2^2 \cdot 17^1 \)
- Exponential form: \( 2^2 \cdot 17 \)
Answer: \( 2^2 \cdot 17 \)
---
- 55 is not divisible by 2, so try the next prime number, 3:
- 55 is not divisible by 3.
- Try the next prime number, 5:
- \( 55 \div 5 = 11 \)
- 11 is a prime number.
- Prime factorization: \( 55 = 5^1 \cdot 11^1 \)
- Exponential form: \( 5 \cdot 11 \)
Answer: \( 5 \cdot 11 \)
---
- Start with 2:
- \( 30 \div 2 = 15 \)
- Now, 15 is not divisible by 2, so try 3:
- \( 15 \div 3 = 5 \)
- 5 is a prime number.
- Prime factorization: \( 30 = 2^1 \cdot 3^1 \cdot 5^1 \)
- Exponential form: \( 2 \cdot 3 \cdot 5 \)
Answer: \( 2 \cdot 3 \cdot 5 \)
---
- Start with 2:
- \( 74 \div 2 = 37 \)
- 37 is a prime number.
- Prime factorization: \( 74 = 2^1 \cdot 37^1 \)
- Exponential form: \( 2 \cdot 37 \)
Answer: \( 2 \cdot 37 \)
---
- 105 is not divisible by 2, so try 3:
- \( 105 \div 3 = 35 \)
- Now, 35 is not divisible by 3, so try 5:
- \( 35 \div 5 = 7 \)
- 7 is a prime number.
- Prime factorization: \( 105 = 3^1 \cdot 5^1 \cdot 7^1 \)
- Exponential form: \( 3 \cdot 5 \cdot 7 \)
Answer: \( 3 \cdot 5 \cdot 7 \)
---
- Start with 2:
- \( 198 \div 2 = 99 \)
- Now, 99 is not divisible by 2, so try 3:
- \( 99 \div 3 = 33 \)
- \( 33 \div 3 = 11 \)
- 11 is a prime number.
- Prime factorization: \( 198 = 2^1 \cdot 3^2 \cdot 11^1 \)
- Exponential form: \( 2 \cdot 3^2 \cdot 11 \)
Answer: \( 2 \cdot 3^2 \cdot 11 \)
---
- Start with 2:
- \( 90 \div 2 = 45 \)
- Now, 45 is not divisible by 2, so try 3:
- \( 45 \div 3 = 15 \)
- \( 15 \div 3 = 5 \)
- 5 is a prime number.
- Prime factorization: \( 90 = 2^1 \cdot 3^2 \cdot 5^1 \)
- Exponential form: \( 2 \cdot 3^2 \cdot 5 \)
Answer: \( 2 \cdot 3^2 \cdot 5 \)
---
- Start with 2:
- \( 48 \div 2 = 24 \)
- \( 24 \div 2 = 12 \)
- \( 12 \div 2 = 6 \)
- \( 6 \div 2 = 3 \)
- 3 is a prime number.
- Prime factorization: \( 48 = 2^4 \cdot 3^1 \)
- Exponential form: \( 2^4 \cdot 3 \)
Answer: \( 2^4 \cdot 3 \)
---
- Start with 2:
- \( 66 \div 2 = 33 \)
- Now, 33 is not divisible by 2, so try 3:
- \( 33 \div 3 = 11 \)
- 11 is a prime number.
- Prime factorization: \( 66 = 2^1 \cdot 3^1 \cdot 11^1 \)
- Exponential form: \( 2 \cdot 3 \cdot 11 \)
Answer: \( 2 \cdot 3 \cdot 11 \)
---
- This is the same as part (c).
- Prime factorization: \( 68 = 2^2 \cdot 17^1 \)
- Exponential form: \( 2^2 \cdot 17 \)
Answer: \( 2^2 \cdot 17 \)
---
- Start with 2:
- \( 42 \div 2 = 21 \)
- Now, 21 is not divisible by 2, so try 3:
- \( 21 \div 3 = 7 \)
- 7 is a prime number.
- Prime factorization: \( 42 = 2^1 \cdot 3^1 \cdot 7^1 \)
- Exponential form: \( 2 \cdot 3 \cdot 7 \)
Answer: \( 2 \cdot 3 \cdot 7 \)
---
- 39 is not divisible by 2, so try 3:
- \( 39 \div 3 = 13 \)
- 13 is a prime number.
- Prime factorization: \( 39 = 3^1 \cdot 13^1 \)
- Exponential form: \( 3 \cdot 13 \)
Answer: \( 3 \cdot 13 \)
---
- 21 is not divisible by 2, so try 3:
- \( 21 \div 3 = 7 \)
- 7 is a prime number.
- Prime factorization: \( 21 = 3^1 \cdot 7^1 \)
- Exponential form: \( 3 \cdot 7 \)
Answer: \( 3 \cdot 7 \)
---
\[
\boxed{
\begin{array}{ccc}
\text{a)} & 2^2 \cdot 5 & \\
\text{b)} & 2^3 \cdot 3^2 & \\
\text{c)} & 2^2 \cdot 17 & \\
\text{d)} & 5 \cdot 11 & \\
\text{e)} & 2 \cdot 3 \cdot 5 & \\
\text{f)} & 2 \cdot 37 & \\
\text{g)} & 3 \cdot 5 \cdot 7 & \\
\text{h)} & 2 \cdot 3^2 \cdot 11 & \\
\text{i)} & 2 \cdot 3^2 \cdot 5 & \\
\text{j)} & 2^4 \cdot 3 & \\
\text{k)} & 2 \cdot 3 \cdot 11 & \\
\text{l)} & 2^2 \cdot 17 & \\
\text{m)} & 2 \cdot 3 \cdot 7 & \\
\text{n)} & 3 \cdot 13 & \\
\text{o)} & 3 \cdot 7 & \\
\end{array}
}
\]
---
a) 20
- Start with the smallest prime number, 2:
- \( 20 \div 2 = 10 \)
- \( 10 \div 2 = 5 \)
- 5 is a prime number.
- Prime factorization: \( 20 = 2^2 \cdot 5^1 \)
- Exponential form: \( 2^2 \cdot 5 \)
Answer: \( 2^2 \cdot 5 \)
---
b) 72
- Start with 2:
- \( 72 \div 2 = 36 \)
- \( 36 \div 2 = 18 \)
- \( 18 \div 2 = 9 \)
- Now, 9 is not divisible by 2, so move to the next prime number, 3:
- \( 9 \div 3 = 3 \)
- \( 3 \div 3 = 1 \)
- Prime factorization: \( 72 = 2^3 \cdot 3^2 \)
- Exponential form: \( 2^3 \cdot 3^2 \)
Answer: \( 2^3 \cdot 3^2 \)
---
c) 68
- Start with 2:
- \( 68 \div 2 = 34 \)
- \( 34 \div 2 = 17 \)
- 17 is a prime number.
- Prime factorization: \( 68 = 2^2 \cdot 17^1 \)
- Exponential form: \( 2^2 \cdot 17 \)
Answer: \( 2^2 \cdot 17 \)
---
d) 55
- 55 is not divisible by 2, so try the next prime number, 3:
- 55 is not divisible by 3.
- Try the next prime number, 5:
- \( 55 \div 5 = 11 \)
- 11 is a prime number.
- Prime factorization: \( 55 = 5^1 \cdot 11^1 \)
- Exponential form: \( 5 \cdot 11 \)
Answer: \( 5 \cdot 11 \)
---
e) 30
- Start with 2:
- \( 30 \div 2 = 15 \)
- Now, 15 is not divisible by 2, so try 3:
- \( 15 \div 3 = 5 \)
- 5 is a prime number.
- Prime factorization: \( 30 = 2^1 \cdot 3^1 \cdot 5^1 \)
- Exponential form: \( 2 \cdot 3 \cdot 5 \)
Answer: \( 2 \cdot 3 \cdot 5 \)
---
f) 74
- Start with 2:
- \( 74 \div 2 = 37 \)
- 37 is a prime number.
- Prime factorization: \( 74 = 2^1 \cdot 37^1 \)
- Exponential form: \( 2 \cdot 37 \)
Answer: \( 2 \cdot 37 \)
---
g) 105
- 105 is not divisible by 2, so try 3:
- \( 105 \div 3 = 35 \)
- Now, 35 is not divisible by 3, so try 5:
- \( 35 \div 5 = 7 \)
- 7 is a prime number.
- Prime factorization: \( 105 = 3^1 \cdot 5^1 \cdot 7^1 \)
- Exponential form: \( 3 \cdot 5 \cdot 7 \)
Answer: \( 3 \cdot 5 \cdot 7 \)
---
h) 198
- Start with 2:
- \( 198 \div 2 = 99 \)
- Now, 99 is not divisible by 2, so try 3:
- \( 99 \div 3 = 33 \)
- \( 33 \div 3 = 11 \)
- 11 is a prime number.
- Prime factorization: \( 198 = 2^1 \cdot 3^2 \cdot 11^1 \)
- Exponential form: \( 2 \cdot 3^2 \cdot 11 \)
Answer: \( 2 \cdot 3^2 \cdot 11 \)
---
i) 90
- Start with 2:
- \( 90 \div 2 = 45 \)
- Now, 45 is not divisible by 2, so try 3:
- \( 45 \div 3 = 15 \)
- \( 15 \div 3 = 5 \)
- 5 is a prime number.
- Prime factorization: \( 90 = 2^1 \cdot 3^2 \cdot 5^1 \)
- Exponential form: \( 2 \cdot 3^2 \cdot 5 \)
Answer: \( 2 \cdot 3^2 \cdot 5 \)
---
j) 48
- Start with 2:
- \( 48 \div 2 = 24 \)
- \( 24 \div 2 = 12 \)
- \( 12 \div 2 = 6 \)
- \( 6 \div 2 = 3 \)
- 3 is a prime number.
- Prime factorization: \( 48 = 2^4 \cdot 3^1 \)
- Exponential form: \( 2^4 \cdot 3 \)
Answer: \( 2^4 \cdot 3 \)
---
k) 66
- Start with 2:
- \( 66 \div 2 = 33 \)
- Now, 33 is not divisible by 2, so try 3:
- \( 33 \div 3 = 11 \)
- 11 is a prime number.
- Prime factorization: \( 66 = 2^1 \cdot 3^1 \cdot 11^1 \)
- Exponential form: \( 2 \cdot 3 \cdot 11 \)
Answer: \( 2 \cdot 3 \cdot 11 \)
---
l) 68
- This is the same as part (c).
- Prime factorization: \( 68 = 2^2 \cdot 17^1 \)
- Exponential form: \( 2^2 \cdot 17 \)
Answer: \( 2^2 \cdot 17 \)
---
m) 42
- Start with 2:
- \( 42 \div 2 = 21 \)
- Now, 21 is not divisible by 2, so try 3:
- \( 21 \div 3 = 7 \)
- 7 is a prime number.
- Prime factorization: \( 42 = 2^1 \cdot 3^1 \cdot 7^1 \)
- Exponential form: \( 2 \cdot 3 \cdot 7 \)
Answer: \( 2 \cdot 3 \cdot 7 \)
---
n) 39
- 39 is not divisible by 2, so try 3:
- \( 39 \div 3 = 13 \)
- 13 is a prime number.
- Prime factorization: \( 39 = 3^1 \cdot 13^1 \)
- Exponential form: \( 3 \cdot 13 \)
Answer: \( 3 \cdot 13 \)
---
o) 21
- 21 is not divisible by 2, so try 3:
- \( 21 \div 3 = 7 \)
- 7 is a prime number.
- Prime factorization: \( 21 = 3^1 \cdot 7^1 \)
- Exponential form: \( 3 \cdot 7 \)
Answer: \( 3 \cdot 7 \)
---
Final Answers:
\[
\boxed{
\begin{array}{ccc}
\text{a)} & 2^2 \cdot 5 & \\
\text{b)} & 2^3 \cdot 3^2 & \\
\text{c)} & 2^2 \cdot 17 & \\
\text{d)} & 5 \cdot 11 & \\
\text{e)} & 2 \cdot 3 \cdot 5 & \\
\text{f)} & 2 \cdot 37 & \\
\text{g)} & 3 \cdot 5 \cdot 7 & \\
\text{h)} & 2 \cdot 3^2 \cdot 11 & \\
\text{i)} & 2 \cdot 3^2 \cdot 5 & \\
\text{j)} & 2^4 \cdot 3 & \\
\text{k)} & 2 \cdot 3 \cdot 11 & \\
\text{l)} & 2^2 \cdot 17 & \\
\text{m)} & 2 \cdot 3 \cdot 7 & \\
\text{n)} & 3 \cdot 13 & \\
\text{o)} & 3 \cdot 7 & \\
\end{array}
}
\]
Parent Tip: Review the logic above to help your child master the concept of factors and prime factorization worksheet.
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