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Find the HCF of 144 and 198.

Answer :

To find the Highest Common Factor (HCF) of 144 and 198, we can use the prime factorization method or the Euclidean algorithm. Here, I'll explain both methods step-by-step.

Method 1: Prime Factorization

  1. Find the prime factors of 144.

    • Divide 144 by 2:

      [tex]144 \div 2 = 72[/tex]

    • Divide 72 by 2:

      [tex]72 \div 2 = 36[/tex]

    • Divide 36 by 2:

      [tex]36 \div 2 = 18[/tex]

    • Divide 18 by 2:

      [tex]18 \div 2 = 9[/tex]

    • Divide 9 by 3:

      [tex]9 \div 3 = 3[/tex]

    • Divide 3 by 3:

      [tex]3 \div 3 = 1[/tex]

    The prime factors of 144 are [tex]2^4 \times 3^2[/tex].

  2. Find the prime factors of 198.

    • Divide 198 by 2:

      [tex]198 \div 2 = 99[/tex]

    • Divide 99 by 3:

      [tex]99 \div 3 = 33[/tex]

    • Divide 33 by 3:

      [tex]33 \div 3 = 11[/tex]

    • Divide 11 by 11:

      [tex]11 \div 11 = 1[/tex]

    The prime factors of 198 are [tex]2^1 \times 3^2 \times 11^1[/tex].

  3. Identify the common prime factors and choose the smallest power for each common factor.

    The common prime factors are 2 and 3:

    • For 2, the smallest power is [tex]2^1[/tex].
    • For 3, the smallest power is [tex]3^2[/tex].

  4. Multiply the common factors:

    [tex]HCF = 2^1 \times 3^2 = 2 \times 9 = 18[/tex].

Method 2: Euclidean Algorithm

  1. Apply the Euclidean algorithm:

    Start with the larger number and divide it by the smaller number:

    [tex]198 \div 144 = 1[/tex] remainder [tex]54[/tex].

  2. Use the remainder as the new divisor:

    [tex]144 \div 54 = 2[/tex] remainder [tex]36[/tex].

  3. Continue with the process:

    [tex]54 \div 36 = 1[/tex] remainder [tex]18[/tex].

    [tex]36 \div 18 = 2[/tex] remainder [tex]0[/tex].

    Since the remainder is 0, the last non-zero remainder, which is 18, is the HCF.

So, the HCF of 144 and 198 is 18.

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