Finding the HCF of 275 and 308

The question asks us to find the Highest Common Factor (HCF) of the numbers 275 and 308. The HCF of two numbers is the largest positive integer that divides both numbers without leaving a remainder.

We can find the HCF using the prime factorization method. This involves breaking down each number into its prime factors.

Prime Factorization Method for HCF

Let's find the prime factors for each number:

Prime Factors of 275:

• Divide 275 by the smallest prime number that divides it. 275 is not divisible by 2 or 3. It is divisible by 5.
• \(\frac{275}{5} = 55\)
• Now, find the prime factors of 55. 55 is divisible by 5.
• \(\frac{55}{5} = 11\)
• 11 is a prime number.
• So, the prime factorization of 275 is \(5 \times 5 \times 11\), which can be written as \(5^2 \times 11^1\).

Prime Factors of 308:

• Divide 308 by the smallest prime number that divides it. 308 is divisible by 2.
• \(\frac{308}{2} = 154\)
• Now, find the prime factors of 154. 154 is divisible by 2.
• \(\frac{154}{2} = 77\)
• Now, find the prime factors of 77. 77 is not divisible by 2, 3, or 5. It is divisible by 7.
• \(\frac{77}{7} = 11\)
• 11 is a prime number.
• So, the prime factorization of 308 is \(2 \times 2 \times 7 \times 11\), which can be written as \(2^2 \times 7^1 \times 11^1\).

Identifying Common Factors

Now, we list the prime factors of both 275 and 308:

• Prime factors of 275: \(5^2 \times 11^1\) (\(5, 5, 11\))
• Prime factors of 308: \(2^2 \times 7^1 \times 11^1\) (\(2, 2, 7, 11\))

To find the HCF, we look for the prime factors that are common to both lists. In this case, the only common prime factor is 11.

For each common prime factor, we take the lowest power that appears in either factorization. The power of 11 in both factorizations is 1 (\(11^1\)).

Calculating the HCF

The HCF is the product of the common prime factors raised to their lowest powers.

Common prime factor: 11

Lowest power of 11: 1

HCF = \(11^1 = 11\)

Therefore, the HCF of 275 and 308 is 11.

This means that 11 is the largest number that can divide both 275 and 308 exactly.

• \(275 \div 11 = 25\)
• \(308 \div 11 = 28\)

Since 25 and 28 have no common factors other than 1, the HCF calculation is correct.

Revision Table: Prime Factors

Additional Information on HCF and Related Concepts

Understanding HCF is important in mathematics. Here are some related concepts:

• Greatest Common Divisor (GCD): HCF is also known as GCD. They are the same concept.
• Euclidean Algorithm: Another method to find the HCF of two numbers, especially useful for larger numbers. It involves repeatedly applying the division lemma until the remainder is 0. The last non-zero remainder is the HCF.
• Least Common Multiple (LCM): The LCM of two numbers is the smallest positive integer that is a multiple of both numbers. There is a relationship between HCF and LCM: For two positive integers a and b, \(HCF(a, b) \times LCM(a, b) = a \times b\).

Finding the HCF helps in simplifying fractions and solving problems involving distribution into equal groups.