Use the prime factor trees to find the highest common factor (HCF) of 330 and 550
The Correct Answer and Explanation is:
To find the highest common factor (HCF) of 330 and 550 using prime factor trees, follow these steps:
Step 1: Prime Factor Tree for 330
Start breaking down 330:
- 330 is divisible by 2:
330 ÷ 2 = 165 - 165 is divisible by 3:
165 ÷ 3 = 55 - 55 is divisible by 5:
55 ÷ 5 = 11 - 11 is a prime number
So, the prime factorization of 330 is:
2 × 3 × 5 × 11
Step 2: Prime Factor Tree for 550
Now break down 550:
- 550 is divisible by 2:
550 ÷ 2 = 275 - 275 is divisible by 5:
275 ÷ 5 = 55 - 55 is divisible by 5:
55 ÷ 5 = 11 - 11 is a prime number
So, the prime factorization of 550 is:
2 × 5 × 5 × 11
Step 3: Identify Common Prime Factors
Now compare the prime factors of both numbers:
- 330: 2, 3, 5, 11
- 550: 2, 5, 5, 11
The common prime factors are:
2, 5, and 11
We only include each common prime factor once when calculating the HCF. Multiply the common factors:
HCF = 2 × 5 × 11 = 110
Final Answer: 110
Explanation
The highest common factor (HCF) of two numbers is the largest number that divides both without leaving a remainder. One efficient method of finding the HCF is using prime factor trees. This method involves breaking down each number into its prime factors then identifying which factors are shared.
For 330, we divide by the smallest prime number that fits. Since 330 is even, we divide by 2. That gives 165. Continuing, we divide 165 by 3 which gives 55. Next, 55 is divisible by 5 and that leaves 11 which is already prime. So the full factorization is 2, 3, 5, and 11.
Next, for 550, we again start with 2 since it’s even. That gives 275. Then we divide by 5 to get 55, and again by 5 to get 11. So the factorization is 2, 5, 5, and 11.
Now we compare the prime factors of both numbers. The numbers 2, 5, and 11 appear in both lists. Even though 550 has two 5s, we only take one for the purpose of finding the HCF. The HCF includes only the prime numbers that appear in both factorizations and only in the smallest power they appear in both.
So multiplying the common factors — 2, 5, and 11 — gives 110. This is the largest number that evenly divides both 330 and 550. Thus, the HCF of 330 and 550 is 110.
