Finding HCF of 310, 208, and 180

The question asks us to find the Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), of the three numbers: 310, 208, and 180.

The HCF is the largest positive integer that divides each of the given numbers without leaving a remainder.

Method: Prime Factorization

One common method to find the HCF is using prime factorization. We find the prime factors of each number separately.

Prime Factorization of 310

Start dividing 310 by the smallest prime number, which is 2:

• \(310 \div 2 = 155\)
• Now, check if 155 is divisible by the next prime numbers. It's not divisible by 3 (1+5+5=11). It is divisible by 5:
• \(155 \div 5 = 31\)
• 31 is a prime number.

So, the prime factorization of 310 is \(2 \times 5 \times 31\).

Prime Factorization of 208

Start dividing 208 by 2:

• \(208 \div 2 = 104\)
• \(104 \div 2 = 52\)
• \(52 \div 2 = 26\)
• \(26 \div 2 = 13\)
• 13 is a prime number.

So, the prime factorization of 208 is \(2 \times 2 \times 2 \times 2 \times 13\), which can be written as \(2^4 \times 13\).

Prime Factorization of 180

Start dividing 180 by 2:

• \(180 \div 2 = 90\)
• \(90 \div 2 = 45\)
• 45 is not divisible by 2. Check the next prime number, 3:
• \(45 \div 3 = 15\)
• \(15 \div 3 = 5\)
• 5 is a prime number.

So, the prime factorization of 180 is \(2 \times 2 \times 3 \times 3 \times 5\), which can be written as \(2^2 \times 3^2 \times 5\).

Identifying Common Factors

Now, let's list the prime factorizations side-by-side:

• \(310 = 2 \times 5 \times 31\)
• \(208 = 2^4 \times 13\)
• \(180 = 2^2 \times 3^2 \times 5\)

We need to find the prime factors that are common to all three numbers. The only prime factor that appears in all three lists is 2.

Calculating the HCF

To find the HCF, we take the lowest power of each common prime factor.

• The prime factor 2 appears in the factorizations as \(2^1\) (in 310), \(2^4\) (in 208), and \(2^2\) (in 180).
• The lowest power of 2 among these is \(2^1\).
• There are no other common prime factors.

Therefore, the HCF is \(2^1 = 2\).

Conclusion

The Highest Common Factor (HCF) of 310, 208, and 180 is 2.