Prime Factorization worksheet for students to find prime factors of numbers and write them in exponential form.

Let’s solve each problem step by step. We need to find the prime factorization of each number and write it in exponential form (like 2² · 5¹ for 20).

We’ll go one by one:

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b) 72

Break down 72:
- 72 2 = 36
- 36 ÷ 2 = 18
- 18 ÷ 2 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1

So primes: 2, 2, 2, 3, 3 → 2³ · 3²

✔ Answer: 2³ · 3²

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c) 68

Break down 68:
- 68 ÷ 2 = 34
- 34 ÷ 2 = 17
- 17 is prime

So primes: 2, 2, 17 → 2² · 17¹

✔ Answer: 2² · 17¹

*(Note: This appears again as “l)” — we’ll do that later too.)*

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d) 55

Break down 55:
- 55 ÷ 5 = 11
- 11 is prime

So primes: 5, 11 → 5¹ · 11¹

✔ Answer: 5¹ · 11¹

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e) 30

Break down 30:
- 30 ÷ 2 = 15
- 15 3 = 5
- 5 is prime

So primes: 2, 3, 5 → 2¹ · 3¹ · 5¹

✔ Answer: 2¹ · 3¹ · 5¹

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f) 74

Break down 74:
- 74 ÷ 2 = 37
- 37 is prime

So primes: 2, 37 → 2¹ · 37¹

✔ Answer: 2¹ · 37¹

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g) 105

Break down 105:
- 105 ÷ 3 = 35
- 35 ÷ 5 = 7
- 7 is prime

So primes: 3, 5, 7 → 3¹ · 5¹ · 7¹

✔ Answer: 3¹ · 5¹ · 7¹

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h) 198

Break down 198:
- 198 ÷ 2 = 99
- 99 3 = 33
- 33 ÷ 3 = 11
- 11 is prime

So primes: 2, 3, 3, 11 → 2¹ · 3² · 11¹

✔ Answer: 2¹ · 3² · 11¹

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i) 90

Break down 90:
- 90 ÷ 2 = 45
- 45 ÷ 3 = 15
- 15 ÷ 3 = 5
- 5 is prime

So primes: 2, 3, 3, 5 → 2¹ · 3² · 5¹

✔ Answer: 2¹ · 3² · 5¹

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j) 48

Break down 48:
- 48 ÷ 2 = 24
- 24 ÷ 2 = 12
- 12 ÷ 2 = 6
- 6 ÷ 2 = 3
- 3 is prime

So primes: 2, 2, 2, 2, 3 → 2 · 3¹

✔ Answer: 2⁴ · 3¹

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k) 66

Break down 66:
- 66 ÷ 2 = 33
- 33 ÷ 3 = 11
- 11 is prime

So primes: 2, 3, 11 → 2¹ · 3¹ · 11¹

✔ Answer: 2¹ · 3¹ · 11¹

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l) 68 ← same as c)

Already did this: 2² · 17¹

✔ Answer: 2² · 17¹

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m) 42

Break down 42:
- 42 ÷ 2 = 21
- 21 ÷ 3 = 7
- 7 is prime

So primes: 2, 3, 7 → 2¹ · 3¹ · 7¹

✔ Answer: 2¹ · 3¹ · 7¹

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n) 39

Break down 39:
- 39 3 = 13
- 13 is prime

So primes: 3, 13 → 3¹ · 13¹

✔ Answer: 3¹ · 13¹

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o) 21

Break down 21:
- 21 ÷ 3 = 7
- 7 is prime

So primes: 3, 7 → 3¹ · 7¹

✔ Answer: 3¹ · 7¹

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Now let’s compile all answers clearly.

Final Answer:
a) 2² · 5¹
b) 2³ · 3²
c) 2² · 17¹
d) 5¹ · 11¹
e) 2¹ · 3¹ · 5¹
f) 2¹ · 37¹
g) 3¹ · 5¹ · 7¹
h) 2¹ · 3² · 11¹
i) 2¹ · 3² · 5¹
j) 2⁴ · 3¹
k) 2¹ · 3¹ · 11¹
l) 2² · 17¹
m) 2¹ · 3¹ · 7¹
n) 3¹ · 13¹
o) 3¹ · 7¹