Find the hcf and lcm of 714 and 135 by prime factorization method
The Correct Answer and Explanation is:
To find the HCF (Highest Common Factor) and LCM (Least Common Multiple) of 714 and 135 using the prime factorization method, follow these steps:
Step 1: Prime Factorization
714:
714 is even, so divide by 2
714 ÷ 2 = 357
357 is divisible by 3
357 ÷ 3 = 119
119 is divisible by 7
119 ÷ 7 = 17
17 is a prime number.
So,
714 = 2 × 3 × 7 × 17
135:
135 is divisible by 3
135 ÷ 3 = 45
45 ÷ 3 = 15
15 ÷ 3 = 5
So,
135 = 3 × 3 × 3 × 5 = 3³ × 5
Step 2: HCF (Highest Common Factor)
Take the common prime factors with the lowest powers.
Both 714 and 135 have only one prime number in common: 3
So,
HCF = 3
Step 3: LCM (Least Common Multiple)
Take all the prime factors from both numbers, using the highest powers:
From 714: 2¹, 3¹, 7¹, 17¹
From 135: 3³, 5¹
So,
LCM = 2 × 3³ × 5 × 7 × 17
Now calculate:
3³ = 27
27 × 2 = 54
54 × 5 = 270
270 × 7 = 1890
1890 × 17 = 32130
So,
LCM = 32130
Final Answers:
- HCF = 3
- LCM = 32130
Explanation
To find the HCF and LCM of two numbers, the prime factorization method is a systematic way that breaks down each number into a product of prime numbers. This allows us to identify common factors and combine factors efficiently.
Starting with 714, we divide by the smallest prime number 2, since it is even. Continuing with 3, 7, and 17, we end up with 2 × 3 × 7 × 17. These are the prime numbers whose product is 714. For 135, we divide by 3 until it is no longer divisible, and then use 5. This gives 3³ × 5.
For the HCF, we look at what prime factors appear in both factorizations. The only common factor is 3, and since 714 has just one 3, we take the lowest power. So, the HCF is 3. This tells us that 3 is the largest number that can divide both 714 and 135 without leaving a remainder.
To find the LCM, we take all prime factors involved in both numbers, but we take the highest power of each. That includes 2¹, 3³, 5¹, 7¹, and 17¹. Multiplying these gives the smallest number that is a multiple of both original numbers.
Using the prime factorization method ensures accuracy and gives a clear view of the relationship between the two numbers. It is particularly useful in mathematics, engineering, and computer science, where factorization plays a key role in algorithms and problem solving.
