Find the LCM of 45 and 75. use their LCM to obtain the HCF
The Correct Answer and Explanation is:
Answer:
- LCM of 45 and 75 = 225
- HCF of 45 and 75 = 15
Step-by-Step Solution:
1. Prime Factorization
- 45 = 3 × 3 × 5 = 3² × 5
- 75 = 3 × 5 × 5 = 3 × 5²
2. Find the LCM (Least Common Multiple)
To find the LCM, take the highest powers of all prime factors involved.
- LCM = 3² × 5² = 9 × 25 = 225
3. Use the relationship between LCM and HCF
There is a key identity that relates LCM and HCF:LCM×HCF=Product of the two numbers\text{LCM} \times \text{HCF} = \text{Product of the two numbers}LCM×HCF=Product of the two numbers
Let’s check:
- 45 × 75 = 3375
- LCM = 225 (from step 2)
So,HCF=45×75LCM=3375225=15\text{HCF} = \frac{45 \times 75}{\text{LCM}} = \frac{3375}{225} = 15HCF=LCM45×75=2253375=15
✅ Hence, the HCF of 45 and 75 is 15.
Explanation
To find the Least Common Multiple (LCM) of two numbers, you look for the smallest number that both numbers divide into without leaving a remainder. The most efficient way to find the LCM is by using prime factorization. For 45, the prime factors are 3² × 5, and for 75, the prime factors are 3 × 5². To get the LCM, you must take the highest powers of all the prime factors that appear in either number. From both numbers, the highest powers are 3² and 5². Multiplying these gives the LCM: 3² × 5² = 9 × 25 = 225.
The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), is the largest number that divides both numbers without leaving a remainder. A key mathematical relationship connects the LCM and HCF of any two positive integers:LCM×HCF=Product of the two numbers\text{LCM} \times \text{HCF} = \text{Product of the two numbers}LCM×HCF=Product of the two numbers
Using this relationship, you can derive the HCF from the LCM. Since we already know that the product of 45 and 75 is 3375 and the LCM is 225, dividing the product by the LCM gives the HCF:3375÷225=153375 \div 225 = 153375÷225=15
This identity is helpful because once you find either the LCM or HCF using one method (such as prime factorization), you can derive the other. This approach is especially useful for large numbers, where direct factoring might be more challenging.
