Printable math worksheet for practicing finding the least common multiple of given number pairs and triples.
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Worksheet titled "Least Common Multiple" with 25 problems asking to find the LCM of various number sets, featuring a cartoon character in the top right corner.
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Step-by-step solution for: Least Common Multiple (LCM) Worksheets
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Show Answer Key & Explanations
Step-by-step solution for: Least Common Multiple (LCM) Worksheets
To solve the problems of finding the Least Common Multiple (LCM) for each set of numbers, we will use the prime factorization method. Here's how it works:
1. Prime Factorization: Break down each number into its prime factors.
2. Identify All Primes: List all the prime numbers that appear in any of the factorizations.
3. Highest Power: For each prime number, take the highest power that appears in any of the factorizations.
4. Multiply: Multiply these highest powers together to get the LCM.
Let’s solve each problem step by step.
---
- Prime Factorization:
- \( 8 = 2^3 \)
- \( 10 = 2 \times 5 \)
- \( 12 = 2^2 \times 3 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^3 \) (from 8)
- \( 3^1 \) (from 12)
- \( 5^1 \) (from 10)
- LCM: \( 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120 \)
Answer: \( \boxed{120} \)
---
- Prime Factorization:
- \( 10 = 2 \times 5 \)
- \( 6 = 2 \times 3 \)
- \( 8 = 2^3 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^3 \) (from 8)
- \( 3^1 \) (from 6)
- \( 5^1 \) (from 10)
- LCM: \( 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120 \)
Answer: \( \boxed{120} \)
---
- Prime Factorization:
- \( 20 = 2^2 \times 5 \)
- \( 15 = 3 \times 5 \)
- \( 10 = 2 \times 5 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^2 \) (from 20)
- \( 3^1 \) (from 15)
- \( 5^1 \) (from 20, 15, and 10)
- LCM: \( 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60 \)
Answer: \( \boxed{60} \)
---
- Prime Factorization:
- \( 24 = 2^3 \times 3 \)
- \( 12 = 2^2 \times 3 \)
- \( 6 = 2 \times 3 \)
- Primes: \( 2, 3 \)
- Highest Powers:
- \( 2^3 \) (from 24)
- \( 3^1 \) (from 24, 12, and 6)
- LCM: \( 2^3 \times 3^1 = 8 \times 3 = 24 \)
Answer: \( \boxed{24} \)
---
- Prime Factorization:
- \( 4 = 2^2 \)
- \( 12 = 2^2 \times 3 \)
- \( 22 = 2 \times 11 \)
- Primes: \( 2, 3, 11 \)
- Highest Powers:
- \( 2^2 \) (from 4 and 12)
- \( 3^1 \) (from 12)
- \( 11^1 \) (from 22)
- LCM: \( 2^2 \times 3^1 \times 11^1 = 4 \times 3 \times 11 = 132 \)
Answer: \( \boxed{132} \)
---
- Prime Factorization:
- \( 12 = 2^2 \times 3 \)
- \( 15 = 3 \times 5 \)
- \( 18 = 2 \times 3^2 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^2 \) (from 12)
- \( 3^2 \) (from 18)
- \( 5^1 \) (from 15)
- LCM: \( 2^2 \times 3^2 \times 5^1 = 4 \times 9 \times 5 = 180 \)
Answer: \( \boxed{180} \)
---
- Prime Factorization:
- \( 50 = 2 \times 5^2 \)
- \( 25 = 5^2 \)
- \( 15 = 3 \times 5 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^1 \) (from 50)
- \( 3^1 \) (from 15)
- \( 5^2 \) (from 50 and 25)
- LCM: \( 2^1 \times 3^1 \times 5^2 = 2 \times 3 \times 25 = 150 \)
Answer: \( \boxed{150} \)
---
- Prime Factorization:
- \( 15 = 3 \times 5 \)
- \( 20 = 2^2 \times 5 \)
- \( 40 = 2^3 \times 5 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^3 \) (from 40)
- \( 3^1 \) (from 15)
- \( 5^1 \) (from 15, 20, and 40)
- LCM: \( 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120 \)
Answer: \( \boxed{120} \)
---
- Prime Factorization:
- \( 12 = 2^2 \times 3 \)
- \( 18 = 2 \times 3^2 \)
- \( 30 = 2 \times 3 \times 5 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^2 \) (from 12)
- \( 3^2 \) (from 18)
- \( 5^1 \) (from 30)
- LCM: \( 2^2 \times 3^2 \times 5^1 = 4 \times 9 \times 5 = 180 \)
Answer: \( \boxed{180} \)
---
- Prime Factorization:
- \( 16 = 2^4 \)
- \( 12 = 2^2 \times 3 \)
- \( 24 = 2^3 \times 3 \)
- Primes: \( 2, 3 \)
- Highest Powers:
- \( 2^4 \) (from 16)
- \( 3^1 \) (from 12 and 24)
- LCM: \( 2^4 \times 3^1 = 16 \times 3 = 48 \)
Answer: \( \boxed{48} \)
---
- Prime Factorization:
- \( 30 = 2 \times 3 \times 5 \)
- \( 15 = 3 \times 5 \)
- \( 75 = 3 \times 5^2 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^1 \) (from 30)
- \( 3^1 \) (from 30, 15, and 75)
- \( 5^2 \) (from 75)
- LCM: \( 2^1 \times 3^1 \times 5^2 = 2 \times 3 \times 25 = 150 \)
Answer: \( \boxed{150} \)
---
- Prime Factorization:
- \( 25 = 5^2 \)
- \( 50 = 2 \times 5^2 \)
- \( 75 = 3 \times 5^2 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^1 \) (from 50)
- \( 3^1 \) (from 75)
- \( 5^2 \) (from 25, 50, and 75)
- LCM: \( 2^1 \times 3^1 \times 5^2 = 2 \times 3 \times 25 = 150 \)
Answer: \( \boxed{150} \)
---
- Prime Factorization:
- \( 8 = 2^3 \)
- \( 24 = 2^3 \times 3 \)
- \( 40 = 2^3 \times 5 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^3 \) (from 8, 24, and 40)
- \( 3^1 \) (from 24)
- \( 5^1 \) (from 40)
- LCM: \( 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120 \)
Answer: \( \boxed{120} \)
---
- Prime Factorization:
- \( 5 = 5 \)
- \( 15 = 3 \times 5 \)
- \( 35 = 5 \times 7 \)
- Primes: \( 3, 5, 7 \)
- Highest Powers:
- \( 3^1 \) (from 15)
- \( 5^1 \) (from 5, 15, and 35)
- \( 7^1 \) (from 35)
- LCM: \( 3^1 \times 5^1 \times 7^1 = 3 \times 5 \times 7 = 105 \)
Answer: \( \boxed{105} \)
---
- Prime Factorization:
- \( 12 = 2^2 \times 3 \)
- \( 36 = 2^2 \times 3^2 \)
- \( 50 = 2 \times 5^2 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^2 \) (from 12 and 36)
- \( 3^2 \) (from 36)
- \( 5^2 \) (from 50)
- LCM: \( 2^2 \times 3^2 \times 5^2 = 4 \times 9 \times 25 = 900 \)
Answer: \( \boxed{900} \)
---
- Prime Factorization:
- \( 8 = 2^3 \)
- \( 16 = 2^4 \)
- \( 100 = 2^2 \times 5^2 \)
- Primes: \( 2, 5 \)
- Highest Powers:
- \( 2^4 \) (from 16)
- \( 5^2 \) (from 100)
- LCM: \( 2^4 \times 5^2 = 16 \times 25 = 400 \)
Answer: \( \boxed{400} \)
---
- Prime Factorization:
- \( 7 = 7 \)
- \( 14 = 2 \times 7 \)
- \( 20 = 2^2 \times 5 \)
- Primes: \( 2, 5, 7 \)
- Highest Powers:
- \( 2^2 \) (from 20)
- \( 5^1 \) (from 20)
- \( 7^1 \) (from 7 and 14)
- LCM: \( 2^2 \times 5^1 \times 7^1 = 4 \times 5 \times 7 = 140 \)
Answer: \( \boxed{140} \)
---
- Prime Factorization:
- \( 10 = 2 \times 5 \)
- \( 20 = 2^2 \times 5 \)
- \( 24 = 2^3 \times 3 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^3 \) (from 24)
- \( 3^1 \) (from 24)
- \( 5^1 \) (from 10 and 20)
- LCM: \( 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120 \)
Answer: \( \boxed{120} \)
---
- Prime Factorization:
- \( 42 = 2 \times 3 \times 7 \)
- \( 14 = 2 \times 7 \)
- \( 21 = 3 \times 7 \)
- Primes: \( 2, 3, 7 \)
- Highest Powers:
- \( 2^1 \) (from 42 and 14)
- \( 3^1 \) (from 42 and 21)
- \( 7^1 \) (from 42, 14, and 21)
- LCM: \( 2^1 \times 3^1 \times 7^1 = 2 \times 3 \times 7 = 42 \)
Answer: \( \boxed{42} \)
---
- Prime Factorization:
- \( 18 = 2 \times 3^2 \)
- \( 27 = 3^3 \)
- \( 15 = 3 \times 5 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^1 \) (from 18)
- \( 3^3 \) (from 27)
- \( 5^1 \) (from 15)
- LCM: \( 2^1 \times 3^3 \times 5^1 = 2 \times 27 \times 5 = 270 \)
Answer: \( \boxed{270} \)
---
- Prime Factorization:
- \( 18 = 2 \times 3^2 \)
- \( 27 = 3^3 \)
- \( 15 = 3 \times 5 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^1 \) (from 18)
- \( 3^3 \) (from 27)
- \( 5^1 \) (from 15)
- LCM: \( 2^1 \times 3^3 \times 5^1 = 2 \times 27 \times 5 = 270 \)
Answer: \( \boxed{270} \)
---
- Prime Factorization:
- \( 10 = 2 \times 5 \)
- \( 20 = 2^2 \times 5 \)
- \( 30 = 2 \times 3 \times 5 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^2 \) (from 20)
- \( 3^1 \) (from 30)
- \( 5^1 \) (from 10, 20, and 30)
- LCM: \( 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60 \)
Answer: \( \boxed{60} \)
---
- Prime Factorization:
- \( 12 = 2^2 \times 3 \)
- \( 24 = 2^3 \times 3 \)
- \( 32 = 2^5 \)
- Primes: \( 2, 3 \)
- Highest Powers:
- \( 2^5 \) (from 32)
- \( 3^1 \) (from 12 and 24)
- LCM: \( 2^5 \times 3^1 = 32 \times 3 = 96 \)
Answer: \( \boxed{96} \)
---
- Prime Factorization:
- \( 32 = 2^5 \)
- \( 48 = 2^4 \times 3 \)
- \( 12 = 2^2 \times 3 \)
- Primes: \( 2, 3 \)
- Highest Powers:
- \( 2^5 \) (from 32)
- \( 3^1 \) (from 48 and 12)
- LCM: \( 2^5 \times 3^1 = 32 \times 3 = 96 \)
Answer: \( \boxed{96} \)
---
- Prime Factorization:
- \( 12 = 2^2 \times 3 \)
- \( 14 = 2 \times 7 \)
- \( 16 = 2^4 \)
- Common Factors: The only common prime factor is \( 2 \).
- HCF: \( 2^1 = 2 \)
Answer: \( \boxed{2} \)
---
\[
\boxed{
\begin{array}{ll}
(1) & 120 \\
(2) & 120 \\
(3) & 60 \\
(4) & 24 \\
(5) & 132 \\
(6) & 180 \\
(7) & 150 \\
(8) & 120 \\
(9) & 180 \\
(10) & 48 \\
(11) & 150 \\
(12) & 150 \\
(13) & 120 \\
(14) & 105 \\
(15) & 900 \\
(16) & 400 \\
(17) & 140 \\
(18) & 120 \\
(19) & 42 \\
(20) & 270 \\
(21) & 270 \\
(22) & 60 \\
(23) & 96 \\
(24) & 96 \\
(25) & 2 \\
\end{array}
}
\]
Steps to Find LCM Using Prime Factorization:
1. Prime Factorization: Break down each number into its prime factors.
2. Identify All Primes: List all the prime numbers that appear in any of the factorizations.
3. Highest Power: For each prime number, take the highest power that appears in any of the factorizations.
4. Multiply: Multiply these highest powers together to get the LCM.
Let’s solve each problem step by step.
---
Problem 1: LCM of 8, 10, and 12
- Prime Factorization:
- \( 8 = 2^3 \)
- \( 10 = 2 \times 5 \)
- \( 12 = 2^2 \times 3 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^3 \) (from 8)
- \( 3^1 \) (from 12)
- \( 5^1 \) (from 10)
- LCM: \( 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120 \)
Answer: \( \boxed{120} \)
---
Problem 2: LCM of 10, 6, and 8
- Prime Factorization:
- \( 10 = 2 \times 5 \)
- \( 6 = 2 \times 3 \)
- \( 8 = 2^3 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^3 \) (from 8)
- \( 3^1 \) (from 6)
- \( 5^1 \) (from 10)
- LCM: \( 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120 \)
Answer: \( \boxed{120} \)
---
Problem 3: LCM of 20, 15, and 10
- Prime Factorization:
- \( 20 = 2^2 \times 5 \)
- \( 15 = 3 \times 5 \)
- \( 10 = 2 \times 5 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^2 \) (from 20)
- \( 3^1 \) (from 15)
- \( 5^1 \) (from 20, 15, and 10)
- LCM: \( 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60 \)
Answer: \( \boxed{60} \)
---
Problem 4: LCM of 24, 12, and 6
- Prime Factorization:
- \( 24 = 2^3 \times 3 \)
- \( 12 = 2^2 \times 3 \)
- \( 6 = 2 \times 3 \)
- Primes: \( 2, 3 \)
- Highest Powers:
- \( 2^3 \) (from 24)
- \( 3^1 \) (from 24, 12, and 6)
- LCM: \( 2^3 \times 3^1 = 8 \times 3 = 24 \)
Answer: \( \boxed{24} \)
---
Problem 5: LCM of 4, 12, and 22
- Prime Factorization:
- \( 4 = 2^2 \)
- \( 12 = 2^2 \times 3 \)
- \( 22 = 2 \times 11 \)
- Primes: \( 2, 3, 11 \)
- Highest Powers:
- \( 2^2 \) (from 4 and 12)
- \( 3^1 \) (from 12)
- \( 11^1 \) (from 22)
- LCM: \( 2^2 \times 3^1 \times 11^1 = 4 \times 3 \times 11 = 132 \)
Answer: \( \boxed{132} \)
---
Problem 6: LCM of 12, 15, and 18
- Prime Factorization:
- \( 12 = 2^2 \times 3 \)
- \( 15 = 3 \times 5 \)
- \( 18 = 2 \times 3^2 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^2 \) (from 12)
- \( 3^2 \) (from 18)
- \( 5^1 \) (from 15)
- LCM: \( 2^2 \times 3^2 \times 5^1 = 4 \times 9 \times 5 = 180 \)
Answer: \( \boxed{180} \)
---
Problem 7: LCM of 50, 25, and 15
- Prime Factorization:
- \( 50 = 2 \times 5^2 \)
- \( 25 = 5^2 \)
- \( 15 = 3 \times 5 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^1 \) (from 50)
- \( 3^1 \) (from 15)
- \( 5^2 \) (from 50 and 25)
- LCM: \( 2^1 \times 3^1 \times 5^2 = 2 \times 3 \times 25 = 150 \)
Answer: \( \boxed{150} \)
---
Problem 8: LCM of 15, 20, and 40
- Prime Factorization:
- \( 15 = 3 \times 5 \)
- \( 20 = 2^2 \times 5 \)
- \( 40 = 2^3 \times 5 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^3 \) (from 40)
- \( 3^1 \) (from 15)
- \( 5^1 \) (from 15, 20, and 40)
- LCM: \( 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120 \)
Answer: \( \boxed{120} \)
---
Problem 9: LCM of 12, 18, and 30
- Prime Factorization:
- \( 12 = 2^2 \times 3 \)
- \( 18 = 2 \times 3^2 \)
- \( 30 = 2 \times 3 \times 5 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^2 \) (from 12)
- \( 3^2 \) (from 18)
- \( 5^1 \) (from 30)
- LCM: \( 2^2 \times 3^2 \times 5^1 = 4 \times 9 \times 5 = 180 \)
Answer: \( \boxed{180} \)
---
Problem 10: LCM of 16, 12, and 24
- Prime Factorization:
- \( 16 = 2^4 \)
- \( 12 = 2^2 \times 3 \)
- \( 24 = 2^3 \times 3 \)
- Primes: \( 2, 3 \)
- Highest Powers:
- \( 2^4 \) (from 16)
- \( 3^1 \) (from 12 and 24)
- LCM: \( 2^4 \times 3^1 = 16 \times 3 = 48 \)
Answer: \( \boxed{48} \)
---
Problem 11: LCM of 30, 15, and 75
- Prime Factorization:
- \( 30 = 2 \times 3 \times 5 \)
- \( 15 = 3 \times 5 \)
- \( 75 = 3 \times 5^2 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^1 \) (from 30)
- \( 3^1 \) (from 30, 15, and 75)
- \( 5^2 \) (from 75)
- LCM: \( 2^1 \times 3^1 \times 5^2 = 2 \times 3 \times 25 = 150 \)
Answer: \( \boxed{150} \)
---
Problem 12: LCM of 25, 50, and 75
- Prime Factorization:
- \( 25 = 5^2 \)
- \( 50 = 2 \times 5^2 \)
- \( 75 = 3 \times 5^2 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^1 \) (from 50)
- \( 3^1 \) (from 75)
- \( 5^2 \) (from 25, 50, and 75)
- LCM: \( 2^1 \times 3^1 \times 5^2 = 2 \times 3 \times 25 = 150 \)
Answer: \( \boxed{150} \)
---
Problem 13: LCM of 8, 24, and 40
- Prime Factorization:
- \( 8 = 2^3 \)
- \( 24 = 2^3 \times 3 \)
- \( 40 = 2^3 \times 5 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^3 \) (from 8, 24, and 40)
- \( 3^1 \) (from 24)
- \( 5^1 \) (from 40)
- LCM: \( 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120 \)
Answer: \( \boxed{120} \)
---
Problem 14: LCM of 5, 15, and 35
- Prime Factorization:
- \( 5 = 5 \)
- \( 15 = 3 \times 5 \)
- \( 35 = 5 \times 7 \)
- Primes: \( 3, 5, 7 \)
- Highest Powers:
- \( 3^1 \) (from 15)
- \( 5^1 \) (from 5, 15, and 35)
- \( 7^1 \) (from 35)
- LCM: \( 3^1 \times 5^1 \times 7^1 = 3 \times 5 \times 7 = 105 \)
Answer: \( \boxed{105} \)
---
Problem 15: LCM of 12, 36, and 50
- Prime Factorization:
- \( 12 = 2^2 \times 3 \)
- \( 36 = 2^2 \times 3^2 \)
- \( 50 = 2 \times 5^2 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^2 \) (from 12 and 36)
- \( 3^2 \) (from 36)
- \( 5^2 \) (from 50)
- LCM: \( 2^2 \times 3^2 \times 5^2 = 4 \times 9 \times 25 = 900 \)
Answer: \( \boxed{900} \)
---
Problem 16: LCM of 8, 16, and 100
- Prime Factorization:
- \( 8 = 2^3 \)
- \( 16 = 2^4 \)
- \( 100 = 2^2 \times 5^2 \)
- Primes: \( 2, 5 \)
- Highest Powers:
- \( 2^4 \) (from 16)
- \( 5^2 \) (from 100)
- LCM: \( 2^4 \times 5^2 = 16 \times 25 = 400 \)
Answer: \( \boxed{400} \)
---
Problem 17: LCM of 7, 14, and 20
- Prime Factorization:
- \( 7 = 7 \)
- \( 14 = 2 \times 7 \)
- \( 20 = 2^2 \times 5 \)
- Primes: \( 2, 5, 7 \)
- Highest Powers:
- \( 2^2 \) (from 20)
- \( 5^1 \) (from 20)
- \( 7^1 \) (from 7 and 14)
- LCM: \( 2^2 \times 5^1 \times 7^1 = 4 \times 5 \times 7 = 140 \)
Answer: \( \boxed{140} \)
---
Problem 18: LCM of 10, 20, and 24
- Prime Factorization:
- \( 10 = 2 \times 5 \)
- \( 20 = 2^2 \times 5 \)
- \( 24 = 2^3 \times 3 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^3 \) (from 24)
- \( 3^1 \) (from 24)
- \( 5^1 \) (from 10 and 20)
- LCM: \( 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120 \)
Answer: \( \boxed{120} \)
---
Problem 19: LCM of 42, 14, and 21
- Prime Factorization:
- \( 42 = 2 \times 3 \times 7 \)
- \( 14 = 2 \times 7 \)
- \( 21 = 3 \times 7 \)
- Primes: \( 2, 3, 7 \)
- Highest Powers:
- \( 2^1 \) (from 42 and 14)
- \( 3^1 \) (from 42 and 21)
- \( 7^1 \) (from 42, 14, and 21)
- LCM: \( 2^1 \times 3^1 \times 7^1 = 2 \times 3 \times 7 = 42 \)
Answer: \( \boxed{42} \)
---
Problem 20: LCM of 18, 27, and 15
- Prime Factorization:
- \( 18 = 2 \times 3^2 \)
- \( 27 = 3^3 \)
- \( 15 = 3 \times 5 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^1 \) (from 18)
- \( 3^3 \) (from 27)
- \( 5^1 \) (from 15)
- LCM: \( 2^1 \times 3^3 \times 5^1 = 2 \times 27 \times 5 = 270 \)
Answer: \( \boxed{270} \)
---
Problem 21: LCM of 18, 27, and 15
- Prime Factorization:
- \( 18 = 2 \times 3^2 \)
- \( 27 = 3^3 \)
- \( 15 = 3 \times 5 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^1 \) (from 18)
- \( 3^3 \) (from 27)
- \( 5^1 \) (from 15)
- LCM: \( 2^1 \times 3^3 \times 5^1 = 2 \times 27 \times 5 = 270 \)
Answer: \( \boxed{270} \)
---
Problem 22: LCM of 10, 20, and 30
- Prime Factorization:
- \( 10 = 2 \times 5 \)
- \( 20 = 2^2 \times 5 \)
- \( 30 = 2 \times 3 \times 5 \)
- Primes: \( 2, 3, 5 \)
- Highest Powers:
- \( 2^2 \) (from 20)
- \( 3^1 \) (from 30)
- \( 5^1 \) (from 10, 20, and 30)
- LCM: \( 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60 \)
Answer: \( \boxed{60} \)
---
Problem 23: LCM of 12, 24, and 32
- Prime Factorization:
- \( 12 = 2^2 \times 3 \)
- \( 24 = 2^3 \times 3 \)
- \( 32 = 2^5 \)
- Primes: \( 2, 3 \)
- Highest Powers:
- \( 2^5 \) (from 32)
- \( 3^1 \) (from 12 and 24)
- LCM: \( 2^5 \times 3^1 = 32 \times 3 = 96 \)
Answer: \( \boxed{96} \)
---
Problem 24: LCM of 32, 48, and 12
- Prime Factorization:
- \( 32 = 2^5 \)
- \( 48 = 2^4 \times 3 \)
- \( 12 = 2^2 \times 3 \)
- Primes: \( 2, 3 \)
- Highest Powers:
- \( 2^5 \) (from 32)
- \( 3^1 \) (from 48 and 12)
- LCM: \( 2^5 \times 3^1 = 32 \times 3 = 96 \)
Answer: \( \boxed{96} \)
---
Problem 25: HCF of 12, 14, and 16
- Prime Factorization:
- \( 12 = 2^2 \times 3 \)
- \( 14 = 2 \times 7 \)
- \( 16 = 2^4 \)
- Common Factors: The only common prime factor is \( 2 \).
- HCF: \( 2^1 = 2 \)
Answer: \( \boxed{2} \)
---
Final Answers:
\[
\boxed{
\begin{array}{ll}
(1) & 120 \\
(2) & 120 \\
(3) & 60 \\
(4) & 24 \\
(5) & 132 \\
(6) & 180 \\
(7) & 150 \\
(8) & 120 \\
(9) & 180 \\
(10) & 48 \\
(11) & 150 \\
(12) & 150 \\
(13) & 120 \\
(14) & 105 \\
(15) & 900 \\
(16) & 400 \\
(17) & 140 \\
(18) & 120 \\
(19) & 42 \\
(20) & 270 \\
(21) & 270 \\
(22) & 60 \\
(23) & 96 \\
(24) & 96 \\
(25) & 2 \\
\end{array}
}
\]
Parent Tip: Review the logic above to help your child master the concept of lcm worksheets.
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