High School

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1. Find the HCF of the following numbers by listing common factors:
a. 14 and 16
b. 27 and 35
c. 20 and 30
d. 18, 52 and 64
e. 40, 45 and 80
f. 12, 21 and 63

2. Find the HCF of the following numbers by using factorisation method:
a. 52 and 70
b. 144 and 290
c. 10 and 85
d. 62, 120
e. 18, 54 and 72
f. 14, 56 and 70

3. Find the HCF of the following numbers by using division method:
a. 28, 40
b. 54, 96
c. 30, 84
d. 49, 63
e. 22, 66 and 88
f. 90, 120 and 160

Answer :

To solve the problem of finding the Highest Common Factor (HCF) using different methods, let's go through each step and approach:

  1. Finding HCF by Listing Common Factors:

    • a. 14 and 16:

      List the factors of each number:

      Factors of 14: [tex]1, 2, 7, 14[/tex]

      Factors of 16: [tex]1, 2, 4, 8, 16[/tex]

      Common factors: [tex]1, 2[/tex]

      The HCF is the largest common factor: [tex]\boxed{2}[/tex]

    • b. 27 and 35:

      Factors of 27: [tex]1, 3, 9, 27[/tex]

      Factors of 35: [tex]1, 5, 7, 35[/tex]

      Common factor: [tex]1[/tex]

      The HCF is: [tex]\boxed{1}[/tex]

    • c. 20 and 30:

      Factors of 20: [tex]1, 2, 4, 5, 10, 20[/tex]

      Factors of 30: [tex]1, 2, 3, 5, 6, 10, 15, 30[/tex]

      Common factors: [tex]1, 2, 5, 10[/tex]

      The HCF is: [tex]\boxed{10}[/tex]

    • d. 18, 52 and 64:

      Factors of 18: [tex]1, 2, 3, 6, 9, 18[/tex]

      Factors of 52: [tex]1, 2, 4, 13, 26, 52[/tex]

      Factors of 64: [tex]1, 2, 4, 8, 16, 32, 64[/tex]

      Common factors: [tex]1, 2[/tex]

      The HCF is: [tex]\boxed{2}[/tex]

    • e. 40, 45 and 80:

      Factors of 40: [tex]1, 2, 4, 5, 8, 10, 20, 40[/tex]

      Factors of 45: [tex]1, 3, 5, 9, 15, 45[/tex]

      Factors of 80: [tex]1, 2, 4, 5, 8, 10, 16, 20, 40, 80[/tex]

      Common factors: [tex]1, 5[/tex]

      The HCF is: [tex]\boxed{5}[/tex]

    • f. 12, 21 and 63:

      Factors of 12: [tex]1, 2, 3, 4, 6, 12[/tex]

      Factors of 21: [tex]1, 3, 7, 21[/tex]

      Factors of 63: [tex]1, 3, 9, 21, 63[/tex]

      Common factor: [tex]1, 3[/tex]

      The HCF is: [tex]\boxed{3}[/tex]

  2. Finding HCF Using Factorisation Method:

    • a. 52 and 70:

      Prime factors of 52: [tex]2^2 \times 13[/tex]

      Prime factors of 70: [tex]2 \times 5 \times 7[/tex]

      Common prime factor: [tex]2[/tex]

      The HCF is: [tex]\boxed{2}[/tex]

    • b. 144 and 290:

      Prime factors of 144: [tex]2^4 \times 3^2[/tex]

      Prime factors of 290: [tex]2 \times 5 \times 29[/tex]

      Common prime factor: [tex]2[/tex]

      The HCF is: [tex]\boxed{2}[/tex]

    • c. 10 and 85:

      Prime factors of 10: [tex]2 \times 5[/tex]

      Prime factors of 85: [tex]5 \times 17[/tex]

      Common prime factor: [tex]5[/tex]

      The HCF is: [tex]\boxed{5}[/tex]

    • d. 62 and 120:

      Prime factors of 62: [tex]2 \times 31[/tex]

      Prime factors of 120: [tex]2^3 \times 3 \times 5[/tex]

      Common prime factor: [tex]2[/tex]

      The HCF is: [tex]\boxed{2}[/tex]

    • e. 18, 54 and 72:

      Prime factors of 18: [tex]2 \times 3^2[/tex]

      Prime factors of 54: [tex]2 \times 3^3[/tex]

      Prime factors of 72: [tex]2^3 \times 3^2[/tex]

      Common prime factors: [tex]2 \times 3^2 = 18[/tex]

      The HCF is: [tex]\boxed{18}[/tex]

    • f. 14, 56 and 70:

      Prime factors of 14: [tex]2 \times 7[/tex]

      Prime factors of 56: [tex]2^3 \times 7[/tex]

      Prime factors of 70: [tex]2 \times 5 \times 7[/tex]

      Common prime factors: [tex]2 \times 7 = 14[/tex]

      The HCF is: [tex]\boxed{14}[/tex]

  3. Finding HCF Using Division Method:

    • a. 28, 40:

      1. Divide 40 by 28, the remainder is 12.
      2. Divide 28 by 12, the remainder is 4.
      3. Divide 12 by 4, the remainder is 0.

      The HCF is the divisor when remainder is 0: [tex]\boxed{4}[/tex]

    • b. 54, 96:

      1. Divide 96 by 54, the remainder is 42.
      2. Divide 54 by 42, the remainder is 12.
      3. Divide 42 by 12, the remainder is 6.
      4. Divide 12 by 6, the remainder is 0.

      The HCF is: [tex]\boxed{6}[/tex]

    • c. 30, 84:

      1. Divide 84 by 30, the remainder is 24.
      2. Divide 30 by 24, the remainder is 6.
      3. Divide 24 by 6, the remainder is 0.

      The HCF is: [tex]\boxed{6}[/tex]

    • d. 49, 63:

      1. Divide 63 by 49, the remainder is 14.
      2. Divide 49 by 14, the remainder is 7.
      3. Divide 14 by 7, the remainder is 0.

      The HCF is: [tex]\boxed{7}[/tex]

    • e. 22, 66 and 88:

      First find HCF of 22 and 66:

      1. Divide 66 by 22, the remainder is 0.

      The HCF of 22 and 66 is: [tex]22[/tex]

      Next, find HCF of 22 and 88:

      1. Divide 88 by 22, the remainder is 0.

      The HCF of 22, 66, and 88 is: [tex]\boxed{22}[/tex]

    • f. 90, 120 and 160:

      First find HCF of 90 and 120:

      1. Divide 120 by 90, the remainder is 30.
      2. Divide 90 by 30, the remainder is 0.

      The HCF of 90 and 120 is: [tex]30[/tex]

      Next, find HCF of 30 and 160:

      1. Divide 160 by 30, the remainder is 10.
      2. Divide 30 by 10, the remainder is 0.

      The HCF of 90, 120, and 160 is: [tex]\boxed{10}[/tex]

By following these approaches, you can find the HCF of numbers using listing common factors, factorization, and division methods. Each method gives you the same result, ensuring the answers are correct.

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