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Write the smallest number which is divisible by both 306 and 657.
The smallest number which is divisible by both 306 and 657 is 22338.
The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers. We can find the LCM of 306 and 657 by listing out the multiples of each number until we find a number that is common to both lists. However, this method can be time-consuming for large numbers.
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There is a more efficient way to find the LCM of two numbers using the prime factorization of each number. The prime factorization of a number is the expression of that number as a product of prime numbers. For example, the prime factorization of 12 is 2^2 * 3.
The LCM of two numbers can be found by taking the product of the highest powers of each prime number that appears in the prime factorization of either number. For example, the LCM of 12 and 18 is 2^2 * 3^2 * 5, which is 180.
Steps to solve:
1. Find the prime factorization of each number:
- Prime factorization of 306: 2 * 3 * 51
- Prime factorization of 657: 3 * 223
2. Identify the highest powers of each prime number:
- Prime number 2: Highest power in 306 (1), highest power in 657 (0)
- Prime number 3: Highest power in 306 (1), highest power in 657 (1)
- Prime number 51: Highest power in 306 (1), highest power in 657 (0)
- Prime number 223: Highest power in 306 (0), highest power in 657 (1)
3. Take the product of the highest powers of each prime number:
- LCM = 2^0 * 3^1 * 51^1 * 223^1 = 22338
Answer: The smallest number which is divisible by both 306 and 657 is 22338.
LCM and HCF of Fractions
To find the Lowest Common Multiple (LCM) and Highest Common Factor (HCF) of fractions, you can follow these steps:
-
For LCM:
- Find the LCM of the denominators of the fractions.
- The LCM of the denominators will be the common denominator for the fractions.
-
For HCF:
- Find the HCF of the numerators.
- Divide both the numerator and denominator of each fraction by this HCF.
Let’s illustrate this with an example:
Example: Find the LCM and HCF of the fractions 2/3 and 4/5.
Step 1: Find LCM The denominators are 3 and 5. The LCM of 3 and 5 is 15. So, the common denominator for the fractions is 15.
Step 2: Find HCF The numerators are 2 and 4. The HCF of 2 and 4 is 2.
Now, let’s simplify the fractions:
- For 2/3, when the denominator is 15, the equivalent fraction is (2/3) * (5/5) = 10/15.
- For 4/5, when the denominator is 15, the equivalent fraction is (4/5) * (3/3) = 12/15.
So, the fractions become 10/15 and 12/15.
Thus, the LCM is 15 and the HCF is 2.
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