Find the LCM of (i) 6, 9 (ii) 10, 15 using prime factorization
The Correct Answer and Explanation is:
To find the Least Common Multiple (LCM) of two numbers, we use their prime factorization. The LCM is the product of the highest powers of all prime factors that appear in the factorization of both numbers.
(i) LCM of 6 and 9:
First, we find the prime factorization of both numbers:
- The prime factorization of 6 is: 6=2×36 = 2 \times 36=2×3
- The prime factorization of 9 is: 9=329 = 3^29=32
Now, to find the LCM, we take the highest power of each prime factor:
- For 2, the highest power is 212^121 (since 6 contains 212^121).
- For 3, the highest power is 323^232 (since 9 contains 323^232).
Thus, the LCM is:LCM=21×32=2×9=18LCM = 2^1 \times 3^2 = 2 \times 9 = 18LCM=21×32=2×9=18
So, the LCM of 6 and 9 is 18.
(ii) LCM of 10 and 15:
Now, let’s find the prime factorization of both numbers:
- The prime factorization of 10 is: 10=2×510 = 2 \times 510=2×5
- The prime factorization of 15 is: 15=3×515 = 3 \times 515=3×5
Again, we take the highest power of each prime factor:
- For 2, the highest power is 212^121 (since 10 contains 212^121).
- For 3, the highest power is 313^131 (since 15 contains 313^131).
- For 5, the highest power is 515^151 (both numbers contain 515^151).
Thus, the LCM is:LCM=21×31×51=2×3×5=30LCM = 2^1 \times 3^1 \times 5^1 = 2 \times 3 \times 5 = 30LCM=21×31×51=2×3×5=30
So, the LCM of 10 and 15 is 30.
Summary:
- The LCM of 6 and 9 is 18.
- The LCM of 10 and 15 is 30.
The LCM is important in many areas of mathematics, especially when adding or subtracting fractions, as it helps find a common denominator.
