Least Common Multiples of Numbers to 100 from Prime Factors with ... - Free Printable

Problem Overview:

The task involves determining the Least Common Multiple (LCM) of pairs of numbers using their prime factorizations. The LCM is calculated by taking the highest power of each prime number that appears in the factorizations of the given numbers.

Solution Explanation:

To find the LCM of two numbers using their prime factorizations:
1. Write the prime factorization of each number.
2. Identify all the prime factors involved.
3. For each prime factor, take the highest power that appears in any of the factorizations.
4. Multiply these highest powers together to get the LCM.

Let's solve each problem step by step:

---

#### Problem 1:
- Numbers: 90 and 39
- Prime factorizations:
- \( 90 = 2 \times 3^2 \times 5 \)
- \( 39 = 3 \times 13 \)
- Combine all prime factors:
- Primes involved: \( 2, 3, 5, 13 \)
- Highest powers:
- \( 2^1 \) (from 90)
- \( 3^2 \) (from 90)
- \( 5^1 \) (from 90)
- \( 13^1 \) (from 39)
- LCM:
\[
\text{LCM} = 2^1 \times 3^2 \times 5^1 \times 13^1 = 2 \times 9 \times 5 \times 13 = 1170
\]

#### Problem 2:
- Numbers: 70 and 60
- Prime factorizations:
- \( 70 = 2 \times 5 \times 7 \)
- \( 60 = 2^2 \times 3 \times 5 \)
- Combine all prime factors:
- Primes involved: \( 2, 3, 5, 7 \)
- Highest powers:
- \( 2^2 \) (from 60)
- \( 3^1 \) (from 60)
- \( 5^1 \) (from both)
- \( 7^1 \) (from 70)
- LCM:
\[
\text{LCM} = 2^2 \times 3^1 \times 5^1 \times 7^1 = 4 \times 3 \times 5 \times 7 = 420
\]

#### Problem 3:
- Numbers: 16 and 28
- Prime factorizations:
- \( 16 = 2^4 \)
- \( 28 = 2^2 \times 7 \)
- Combine all prime factors:
- Primes involved: \( 2, 7 \)
- Highest powers:
- \( 2^4 \) (from 16)
- \( 7^1 \) (from 28)
- LCM:
\[
\text{LCM} = 2^4 \times 7^1 = 16 \times 7 = 112
\]

#### Problem 4:
- Numbers: 76 and 86
- Prime factorizations:
- \( 76 = 2^2 \times 19 \)
- \( 86 = 2 \times 43 \)
- Combine all prime factors:
- Primes involved: \( 2, 19, 43 \)
- Highest powers:
- \( 2^2 \) (from 76)
- \( 19^1 \) (from 76)
- \( 43^1 \) (from 86)
- LCM:
\[
\text{LCM} = 2^2 \times 19^1 \times 43^1 = 4 \times 19 \times 43 = 3268
\]

#### Problem 5:
- Numbers: 68 and 58
- Prime factorizations:
- \( 68 = 2^2 \times 17 \)
- \( 58 = 2 \times 29 \)
- Combine all prime factors:
- Primes involved: \( 2, 17, 29 \)
- Highest powers:
- \( 2^2 \) (from 68)
- \( 17^1 \) (from 68)
- \( 29^1 \) (from 58)
- LCM:
\[
\text{LCM} = 2^2 \times 17^1 \times 29^1 = 4 \times 17 \times 29 = 1972
\]

#### Problem 6:
- Numbers: 66 and 10
- Prime factorizations:
- \( 66 = 2 \times 3 \times 11 \)
- \( 10 = 2 \times 5 \)
- Combine all prime factors:
- Primes involved: \( 2, 3, 5, 11 \)
- Highest powers:
- \( 2^1 \) (from both)
- \( 3^1 \) (from 66)
- \( 5^1 \) (from 10)
- \( 11^1 \) (from 66)
- LCM:
\[
\text{LCM} = 2^1 \times 3^1 \times 5^1 \times 11^1 = 2 \times 3 \times 5 \times 11 = 330
\]

#### Problem 7:
- Numbers: 56 and 46
- Prime factorizations:
- \( 56 = 2^3 \times 7 \)
- \( 46 = 2 \times 23 \)
- Combine all prime factors:
- Primes involved: \( 2, 7, 23 \)
- Highest powers:
- \( 2^3 \) (from 56)
- \( 7^1 \) (from 56)
- \( 23^1 \) (from 46)
- LCM:
\[
\text{LCM} = 2^3 \times 7^1 \times 23^1 = 8 \times 7 \times 23 = 1288
\]

#### Problem 8:
- Numbers: 39 and 57
- Prime factorizations:
- \( 39 = 3 \times 13 \)
- \( 57 = 3 \times 19 \)
- Combine all prime factors:
- Primes involved: \( 3, 13, 19 \)
- Highest powers:
- \( 3^1 \) (from both)
- \( 13^1 \) (from 39)
- \( 19^1 \) (from 57)
- LCM:
\[
\text{LCM} = 3^1 \times 13^1 \times 19^1 = 3 \times 13 \times 19 = 741
\]

#### Problem 9:
- Numbers: 20 and 48
- Prime factorizations:
- \( 20 = 2^2 \times 5 \)
- \( 48 = 2^4 \times 3 \)
- Combine all prime factors:
- Primes involved: \( 2, 3, 5 \)
- Highest powers:
- \( 2^4 \) (from 48)
- \( 3^1 \) (from 48)
- \( 5^1 \) (from 20)
- LCM:
\[
\text{LCM} = 2^4 \times 3^1 \times 5^1 = 16 \times 3 \times 5 = 240
\]

#### Problem 10:
- Numbers: 74 and 4
- Prime factorizations:
- \( 74 = 2 \times 37 \)
- \( 4 = 2^2 \)
- Combine all prime factors:
- Primes involved: \( 2, 37 \)
- Highest powers:
- \( 2^2 \) (from 4)
- \( 37^1 \) (from 74)
- LCM:
\[
\text{LCM} = 2^2 \times 37^1 = 4 \times 37 = 148
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \quad 1170 \\
2. & \quad 420 \\
3. & \quad 112 \\
4. & \quad 3268 \\
5. & \quad 1972 \\
6. & \quad 330 \\
7. & \quad 1288 \\
8. & \quad 741 \\
9. & \quad 240 \\
10. & \quad 148 \\
\end{aligned}
}
\]