To find the HCF (Highest Common Factor) of 2616 and 2289, we will use the prime factorization method. Here’s a step-by-step solution:
### Step 1: Prime Factorization of 2616
To find the prime factors of 2616, we can start dividing by the smallest prime numbers.
- 2616 is even, so we divide by 2:
\[
2616 \div 2 = 1308
\]
- 1308 is also even, divide by 2 again:
\[
1308 \div 2 = 654
\]
- 654 is even, divide by 2 again:
\[
654 \div 2 = 327
\]
- Now, 327 is not even, so we check for the next prime number, which is 3:
\[
327 \div 3 = 109
\]
- 109 is a prime number.
Thus, the prime factorization of 2616 is:
\[
2616 = 2^3 \times 3^1 \times 109^1
\]
### Step 2: Prime Factorization of 2289
Now, we will find the prime factors of 2289.
- 2289 is odd, so we start with 3:
\[
2289 \div 3 = 763
\]
- Next, we check if 763 is divisible by 3, 5, or 7. It is not divisible by these, but it is divisible by 109:
\[
763 \div 109 = 7
\]
- 7 is a prime number.
Thus, the prime factorization of 2289 is:
\[
2289 = 3^1 \times 7^1 \times 109^1
\]
### Step 3: Identify Common Prime Factors
Now, we will identify the common prime factors between 2616 and 2289.
- From the factorization:
- 2616: \(2^3, 3^1, 109^1\)
- 2289: \(3^1, 7^1, 109^1\)
The common prime factors are:
- \(3^1\)
- \(109^1\)
### Step 4: Calculate the HCF
To find the HCF, we multiply the common prime factors:
\[
HCF = 3^1 \times 109^1 = 3 \times 109 = 327
\]
### Conclusion
The HCF of 2616 and 2289 is **327**.
---
