To find the highest common factor (HCF) of 360 and 450, we can follow these steps:
### Step 1: Prime Factorization of 360
First, we need to find the prime factors of 360.
1. Divide 360 by 2 (the smallest prime number):
- 360 ÷ 2 = 180
2. Divide 180 by 2:
- 180 ÷ 2 = 90
3. Divide 90 by 2:
- 90 ÷ 2 = 45
4. Now, 45 is not divisible by 2, so we move to the next prime number, which is 3:
- 45 ÷ 3 = 15
5. Divide 15 by 3:
- 15 ÷ 3 = 5
6. Finally, 5 is a prime number.
Thus, the prime factorization of 360 is:
\[ 360 = 2^3 \times 3^2 \times 5^1 \]
### Step 2: Prime Factorization of 450
Next, we find the prime factors of 450.
1. Divide 450 by 2:
- 450 ÷ 2 = 225
2. Now, 225 is not divisible by 2, so we move to the next prime number, which is 3:
- 225 ÷ 3 = 75
3. Divide 75 by 3:
- 75 ÷ 3 = 25
4. Now, 25 is not divisible by 3, so we move to the next prime number, which is 5:
- 25 ÷ 5 = 5
5. Finally, divide 5 by 5:
- 5 ÷ 5 = 1
Thus, the prime factorization of 450 is:
\[ 450 = 2^1 \times 3^2 \times 5^2 \]
### Step 3: Identify Common Factors
Now, we identify the common prime factors from both factorizations:
- From 360: \( 2^3, 3^2, 5^1 \)
- From 450: \( 2^1, 3^2, 5^2 \)
The common prime factors are:
- For 2: The minimum power is \( 2^1 \)
- For 3: The minimum power is \( 3^2 \)
- For 5: The minimum power is \( 5^1 \)
### Step 4: Calculate HCF
Now, we multiply the common prime factors together:
\[ \text{HCF} = 2^1 \times 3^2 \times 5^1 \]
Calculating this:
- \( 2^1 = 2 \)
- \( 3^2 = 9 \)
- \( 5^1 = 5 \)
Now, multiply these together:
\[ \text{HCF} = 2 \times 9 \times 5 = 90 \]
### Conclusion
Thus, the highest common factor (HCF) of 360 and 450 is **90**.
