Factors Worksheets | Printable Factors and Multiples Worksheets - Free Printable

To solve the problems of finding the Least Common Multiple (LCM) and Greatest Common Factor (GCF) using prime factorization, we will follow these steps:

Steps:

1. Prime Factorization: Break down each number into its prime factors.
2. GCF: Identify the common prime factors and take the lowest power of each.
3. LCM: Take the highest power of all the prime factors present in any of the numbers.

Let's solve each problem step by step.

---

Problem 1: 4, 5, 24

#### Prime Factorization:
- \( 4 = 2^2 \)
- \( 5 = 5^1 \)
- \( 24 = 2^3 \times 3^1 \)

#### GCF:
- The only common factor is \( 1 \) (since there are no shared prime factors).
- GCF = 1

#### LCM:
- Take the highest powers of all prime factors: \( 2^3 \), \( 3^1 \), \( 5^1 \).
- \( \text{LCM} = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120 \)
- LCM = 120

Answer:
- LCM: 120
- GCF: 1

---

Problem 2: 4, 15, 16

#### Prime Factorization:
- \( 4 = 2^2 \)
- \( 15 = 3^1 \times 5^1 \)
- \( 16 = 2^4 \)

#### GCF:
- The only common factor is \( 1 \) (since there are no shared prime factors).
- GCF = 1

#### LCM:
- Take the highest powers of all prime factors: \( 2^4 \), \( 3^1 \), \( 5^1 \).
- \( \text{LCM} = 2^4 \times 3^1 \times 5^1 = 16 \times 3 \times 5 = 240 \)
- LCM = 240

Answer:
- LCM: 240
- GCF: 1

---

Problem 3: 32, 34, 36

#### Prime Factorization:
- \( 32 = 2^5 \)
- \( 34 = 2^1 \times 17^1 \)
- \( 36 = 2^2 \times 3^2 \)

#### GCF:
- The only common factor is \( 2^1 \).
- GCF = 2

#### LCM:
- Take the highest powers of all prime factors: \( 2^5 \), \( 3^2 \), \( 17^1 \).
- \( \text{LCM} = 2^5 \times 3^2 \times 17^1 = 32 \times 9 \times 17 = 4896 \)
- LCM = 4896

Answer:
- LCM: 4896
- GCF: 2

---

Problem 4: 34, 35, 32

#### Prime Factorization:
- \( 34 = 2^1 \times 17^1 \)
- \( 35 = 5^1 \times 7^1 \)
- \( 32 = 2^5 \)

#### GCF:
- The only common factor is \( 1 \) (since there are no shared prime factors).
- GCF = 1

#### LCM:
- Take the highest powers of all prime factors: \( 2^5 \), \( 5^1 \), \( 7^1 \), \( 17^1 \).
- \( \text{LCM} = 2^5 \times 5^1 \times 7^1 \times 17^1 = 32 \times 5 \times 7 \times 17 = 19040 \)
- LCM = 19040

Answer:
- LCM: 19040
- GCF: 1

---

Problem 5: 21, 24, 36

#### Prime Factorization:
- \( 21 = 3^1 \times 7^1 \)
- \( 24 = 2^3 \times 3^1 \)
- \( 36 = 2^2 \times 3^2 \)

#### GCF:
- The common factor is \( 3^1 \).
- GCF = 3

#### LCM:
- Take the highest powers of all prime factors: \( 2^3 \), \( 3^2 \), \( 7^1 \).
- \( \text{LCM} = 2^3 \times 3^2 \times 7^1 = 8 \times 9 \times 7 = 504 \)
- LCM = 504

Answer:
- LCM: 504
- GCF: 3

---

Problem 6: 10, 16, 21

#### Prime Factorization:
- \( 10 = 2^1 \times 5^1 \)
- \( 16 = 2^4 \)
- \( 21 = 3^1 \times 7^1 \)

#### GCF:
- The only common factor is \( 1 \) (since there are no shared prime factors).
- GCF = 1

#### LCM:
- Take the highest powers of all prime factors: \( 2^4 \), \( 3^1 \), \( 5^1 \), \( 7^1 \).
- \( \text{LCM} = 2^4 \times 3^1 \times 5^1 \times 7^1 = 16 \times 3 \times 5 \times 7 = 1680 \)
- LCM = 1680

Answer:
- LCM: 1680
- GCF: 1

---

Problem 7: 21, 36, 32

#### Prime Factorization:
- \( 21 = 3^1 \times 7^1 \)
- \( 36 = 2^2 \times 3^2 \)
- \( 32 = 2^5 \)

#### GCF:
- The only common factor is \( 1 \) (since there are no shared prime factors).
- GCF = 1

#### LCM:
- Take the highest powers of all prime factors: \( 2^5 \), \( 3^2 \), \( 7^1 \).
- \( \text{LCM} = 2^5 \times 3^2 \times 7^1 = 32 \times 9 \times 7 = 2016 \)
- LCM = 2016

Answer:
- LCM: 2016
- GCF: 1

---

Problem 8: 34, 21, 14

#### Prime Factorization:
- \( 34 = 2^1 \times 17^1 \)
- \( 21 = 3^1 \times 7^1 \)
- \( 14 = 2^1 \times 7^1 \)

#### GCF:
- The only common factor is \( 1 \) (since there are no shared prime factors other than \( 1 \)).
- GCF = 1

#### LCM:
- Take the highest powers of all prime factors: \( 2^1 \), \( 3^1 \), \( 7^1 \), \( 17^1 \).
- \( \text{LCM} = 2^1 \times 3^1 \times 7^1 \times 17^1 = 2 \times 3 \times 7 \times 17 = 714 \)
- LCM = 714

Answer:
- LCM: 714
- GCF: 1

---

Problem 9: 35, 15, 12

#### Prime Factorization:
- \( 35 = 5^1 \times 7^1 \)
- \( 15 = 3^1 \times 5^1 \)
- \( 12 = 2^2 \times 3^1 \)

#### GCF:
- The only common factor is \( 1 \) (since there are no shared prime factors other than \( 1 \)).
- GCF = 1

#### LCM:
- Take the highest powers of all prime factors: \( 2^2 \), \( 3^1 \), \( 5^1 \), \( 7^1 \).
- \( \text{LCM} = 2^2 \times 3^1 \times 5^1 \times 7^1 = 4 \times 3 \times 5 \times 7 = 420 \)
- LCM = 420

Answer:
- LCM: 420
- GCF: 1

---

Problem 10: 34, 15, 4

#### Prime Factorization:
- \( 34 = 2^1 \times 17^1 \)
- \( 15 = 3^1 \times 5^1 \)
- \( 4 = 2^2 \)

#### GCF:
- The only common factor is \( 1 \) (since there are no shared prime factors).
- GCF = 1

#### LCM:
- Take the highest powers of all prime factors: \( 2^2 \), \( 3^1 \), \( 5^1 \), \( 17^1 \).
- \( \text{LCM} = 2^2 \times 3^1 \times 5^1 \times 17^1 = 4 \times 3 \times 5 \times 17 = 1020 \)
- LCM = 1020

Answer:
- LCM: 1020
- GCF: 1

---

Final Answers:

1. LCM: 120, GCF: 1
2. LCM: 240, GCF: 1
3. LCM: 4896, GCF: 2
4. LCM: 19040, GCF: 1
5. LCM: 504, GCF: 3
6. LCM: 1680, GCF: 1
7. LCM: 2016, GCF: 1
8. LCM: 714, GCF: 1
9. LCM: 420, GCF: 1
10. LCM: 1020, GCF: 1

\boxed{
\begin{array}{ll}
1. & \text{LCM: 120, GCF: 1} \\
2. & \text{LCM: 240, GCF: 1} \\
3. & \text{LCM: 4896, GCF: 2} \\
4. & \text{LCM: 19040, GCF: 1} \\
5. & \text{LCM: 504, GCF: 3} \\
6. & \text{LCM: 1680, GCF: 1} \\
7. & \text{LCM: 2016, GCF: 1} \\
8. & \text{LCM: 714, GCF: 1} \\
9. & \text{LCM: 420, GCF: 1} \\
10. & \text{LCM: 1020, GCF: 1} \\
\end{array}
}