The HCF of the numbers 310, 208, and 180 is:
2
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.
Similar Questions
- Three bells ring at intervals of 15, 30 and 45 minutes respectively. At what time will they ring together again, if they rang simultaneously at 8.00 AM?
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