Finding the HCF of 98, 175, and 210

The problem asks us to find the Highest Common Factor (HCF) of three numbers: 98, 175, and 210.

The HCF of a set of numbers is the largest positive integer that divides each number exactly without leaving a remainder. To find the HCF, we can use the method of prime factorization.

Understanding Prime Factorization

Prime factorization is the process of breaking down a number into its prime factors, which are prime numbers that multiply together to give the original number. Every composite number has a unique prime factorization.

Step-by-Step Prime Factorization

Let's find the prime factors for each of the given numbers:

1. Prime Factorization of 98:

• Divide 98 by the smallest prime number, 2: $98 \div 2 = 49$.
• 49 is not divisible by 2 or 3. The next prime number is 5, which does not divide 49. The next prime number is 7.
• Divide 49 by 7: $49 \div 7 = 7$.
• 7 is a prime number.

So, the prime factorization of 98 is $2 \times 7 \times 7 = 2 \times 7^2$.

2. Prime Factorization of 175:

• 175 is an odd number, so it's not divisible by 2. The sum of its digits is $1+7+5 = 13$, which is not divisible by 3, so 175 is not divisible by 3.
• 175 ends in 5, so it is divisible by 5: $175 \div 5 = 35$.
• 35 ends in 5, so it is divisible by 5: $35 \div 5 = 7$.
• 7 is a prime number.

So, the prime factorization of 175 is $5 \times 5 \times 7 = 5^2 \times 7$.

3. Prime Factorization of 210:

• 210 is an even number, so it's divisible by 2: $210 \div 2 = 105$.
• The sum of the digits of 105 is $1+0+5=6$, which is divisible by 3, so 105 is divisible by 3: $105 \div 3 = 35$.
• 35 is divisible by 5: $35 \div 5 = 7$.
• 7 is a prime number.

So, the prime factorization of 210 is $2 \times 3 \times 5 \times 7$.

Identifying Common Factors for HCF

Now, let's look at the prime factorizations of all three numbers and identify the prime factors that are common to all of them. For the HCF, we take the lowest power of each common prime factor.

• Prime factors of 98: $2^1, 7^2$
• Prime factors of 175: $5^2, 7^1$
• Prime factors of 210: $2^1, 3^1, 5^1, 7^1$

Comparing the prime factors:

• The prime factor 2 appears in 98 and 210, but not in 175. So, 2 is not a common factor for all three.
• The prime factor 3 appears in 210, but not in 98 or 175. So, 3 is not a common factor for all three.
• The prime factor 5 appears in 175 and 210, but not in 98. So, 5 is not a common factor for all three.
• The prime factor 7 appears in 98 ($7^2$), 175 ($7^1$), and 210 ($7^1$). It is a common factor. The lowest power of 7 that appears in all three factorizations is $7^1$.

The only common prime factor is 7, and its lowest power is 1.

Calculating the HCF

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

HCF$(98, 175, 210) = 7^1 = 7$.

Therefore, the HCF of 98, 175, and 210 is 7.

Revision Table: Understanding HCF and Prime Factorization

Additional Information: HCF vs. LCM

The HCF is the largest number that divides the given numbers. A related concept is the Least Common Multiple (LCM), which is the smallest positive integer that is a multiple of two or more numbers.

For example, let's consider 6 and 8:

• Factors of 6: 1, 2, 3, 6
• Factors of 8: 1, 2, 4, 8
• Common factors: 1, 2
• HCF(6, 8) = 2

• Multiples of 6: 6, 12, 18, 24, 30, ...
• Multiples of 8: 8, 16, 24, 32, ...
• Common multiples: 24, 48, ...
• LCM(6, 8) = 24

While HCF is found using the lowest powers of common prime factors, LCM is found using the highest powers of all prime factors involved in the numbers' factorizations.