Finding the HCF of 36 and 198

The problem asks us to find the Highest Common Factor (HCF) of the numbers 36 and 198. The HCF is the largest positive integer that divides two or more numbers without leaving a remainder. We can find the HCF using methods like prime factorization or the Euclidean algorithm. Let's use the prime factorization method as it's quite intuitive for these numbers.

Prime Factorization Method Explained

Prime factorization involves breaking down each number into its prime factors. Prime numbers are numbers greater than 1 that have only two divisors: 1 and themselves (examples: 2, 3, 5, 7, 11, etc.). Once we have the prime factors for both numbers, we identify the common factors and multiply them together, taking the lowest power of each common prime factor.

Step 1: Prime Factorization of 36

Let's find the prime factors of 36:

• Divide 36 by the smallest prime number, 2: $36 \div 2 = 18$
• Divide 18 by 2: $18 \div 2 = 9$
• 9 is not divisible by 2, so try the next prime number, 3: $9 \div 3 = 3$
• Divide 3 by 3: $3 \div 3 = 1$

So, the prime factorization of 36 is $2 \times 2 \times 3 \times 3$, which can be written in exponential form as $2^2 \times 3^2$.

Step 2: Prime Factorization of 198

Now, let's find the prime factors of 198:

• Divide 198 by 2: $198 \div 2 = 99$
• 99 is not divisible by 2, so try 3: $99 \div 3 = 33$
• Divide 33 by 3: $33 \div 3 = 11$
• 11 is a prime number. Divide 11 by 11: $11 \div 11 = 1$

So, the prime factorization of 198 is $2 \times 3 \times 3 \times 11$, which can be written in exponential form as $2^1 \times 3^2 \times 11^1$.

Step 3: Identify Common Prime Factors

Now we compare the prime factorizations of 36 and 198:

Prime factors of $36 = 2^2 \times 3^2$

Prime factors of $198 = 2^1 \times 3^2 \times 11^1$

The common prime factors are 2 and 3.

Step 4: Calculate the HCF

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

• For the prime factor 2, the powers are $2^2$ (from 36) and $2^1$ (from 198). The lowest power is $2^1$.
• For the prime factor 3, the powers are $3^2$ (from 36) and $3^2$ (from 198). The lowest power is $3^2$.
• The prime factor 11 is only present in 198, not common to both.

HCF is the product of these lowest powers of common factors:

HCF = $2^1 \times 3^2 = 2 \times (3 \times 3) = 2 \times 9 = 18$.

Thus, the HCF of 36 and 198 is 18.

Confirming the HCF

We can check if 18 divides both 36 and 198 without a remainder:

• $36 \div 18 = 2$ (Remainder 0)
• $198 \div 18 = 11$ (Remainder 0)

Since 18 divides both numbers perfectly, and based on our calculation, it is the highest such number, the HCF is indeed 18.

Revision Table: Key Concepts

Additional Information: HCF and LCM

Besides HCF, another important concept is the Least Common Multiple (LCM). The LCM is the smallest positive integer that is a multiple of two or more numbers.

While HCF uses the lowest powers of common prime factors, LCM uses the highest powers of all prime factors (common and non-common).

For 36 ($2^2 \times 3^2$) and 198 ($2^1 \times 3^2 \times 11^1$):

• Highest power of 2 is $2^2$.
• Highest power of 3 is $3^2$.
• Highest power of 11 is $11^1$.

LCM(36, 198) = $2^2 \times 3^2 \times 11^1 = 4 \times 9 \times 11 = 36 \times 11 = 396$.

There is a relationship between HCF and LCM for two numbers, A and B:

HCF(A, B) $\times$ LCM(A, B) = A $\times$ B

Let's check this for 36 and 198:

$18 \times 396 = 7128$

$36 \times 198 = 7128$

The relationship holds true. Understanding both HCF and LCM is crucial for number theory problems.