Define HCF of two positive integers and find the HCF of the following pairs of numbers:

(i) 32 and 54 (ii) 18 and 24

(iii) 70 and 30 (iv) 56 and 88

(v) 475 and 495 (vi) 75 and 243

(vii)240 and 6552(viii)155 and 1385

(ix) 100 and 190 (x) 105 and 120

Definition of HCF (Highest Common Factor): The largest positive integer which divides two or more integers without any remainder is called Highest Common Factor (HCF) or Greatest Common Divisor or Greatest Common Factor (GCF).

(i) Prime factorization of 32 and 54 are:

32 = 2 × 2 × 2 × 2 × 2

54 = 2 × 3 × 3 × 3

From above prime factorization we got that the highest common factor of 32 and 54 is 2

(ii) Prime factorization of 18 and 24 are:

18 = 2 × 3 × 3

24 = 2 × 2 × 2 × 3

From above prime factorization we got that the highest common factor of 18 and 24 is 3×2 ⇒ 6

(iii) Prime factorization of 30 and 70 are:

30 = 2 × 3 × 5

70 = 2 × 5 × 7

From above prime factorization we got that the highest common factor of 30 and 70 is 2×5 ⇒ 10

(iv) Prime factorization of 56 and 88 are:

56 = 2 × 2 × 2 × 7

88 = 2 × 2 × 2 × 11

From above prime factorization we got that the highest common factor of 56 and 88 is 2×2×2 ⇒ 8

(v) Prime factorization of 475 and 495 are:

475 = 5 × 5 × 19

495 = 3 × 3 × 5 × 11

From above prime factorization we got that the highest common factor of 475 and 495 is 5

(vi) Prime factorization of 75 and 243 are:

75 = 3 × 5 × 5

243 = 3 × 3 x 3 × 3 × 3

From above prime factorization we got that the highest common factor of 75 and 243 is 3

(vii) Prime factorization of 75 and 243 are:

240 = 2 × 2 ×2 × 2 × 3 × 5

6552 = 2 × 2 × 2 × 3 x 3 × 7 × 13

From above prime factorization we got that the highest common factor of 240 and 6552 is 2×2×2×3 ⇒ 24

(viii) Prime factorization of 155 and 1385 are:

155 = 5 × 31

1385 = 5 × 277

From above prime factorization we got that the highest common factor of 155 and 1385 is 5

(ix) Prime factorization of 100 and 190 are:

100 = 2 × 2 × 5 × 5

190 = 2 × 5 ×19

From above prime factorization we got that the highest common factor of 155 and 1385 is 2×5 ⇒ 10

(x) Prime factorization of 105 and 120 are:

105 = 3 × 5 × 7

120 = 2 × 2 × 2 × 3 × 5

From above prime factorization we got that the highest common factor of 105 and 120 is 3×5 ⇒ 15