Question

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

Answer 1

HCF (highest common factor) :  

The largest positive integer that divides given two positive integers is called the Highest Common Factor of these positive integers.

(i) Given : Two positive integers  32 and 54.

Here, 54 > 32

Let a = 54 and b = 32

54 = 32 x 1 + 22

[By applying division lemma, a = bq + r]

Here, remainder = 22 ≠ 0, so take new dividend as 32 and new divisor as 22.

Let a = 32 and b= 22

32 = 22 x 1 + 10

Here, remainder = 10  ≠ 0, so take new dividend as 22 and new divisor as 10.

Let a = 22 and b= 10

22 = 10 x 2 + 2

Here,remainder = 2 ≠ 0, so take new dividend as 10 and new divisor as 2.

Let a = 10 and b= 2.

10 = 2 x 5 + 0

Here, remainder is zero and divisor is 2.

Hence , H.C.F. of 32 and 54 is 2.

(ii) Given : Two positive integers  18 and 24.

Here, 24 > 18

Let a = 24 and b = 18

24 = 18 x 1 + 6.

[By applying division lemma, a = bq + r]

Here, remainder = 6 ≠ 0, so take new dividend as 18 and new divisor as 6.

Let a = 18 and b= 6

18 = 6 x 3 + 0.

Here, remainder is zero and divisor is 6.

Hence ,H.C.F. of 18 and 24 is 6.

(iii) Given : Two positive integers  70 and 30

Here, 70 > 30

Let a = 70 and b = 30

70 = 30 x 2 + 10.

[By applying division lemma, a = bq + r]

Here, remainder = 10 ≠ 0, so take new dividend as 30 and new divisor as 10.

Let a = 30 and b= 10

30 = 10 x 3 + 0.

Here, remainder is zero and divisor is 10.

Hence ,H.C.F. of 70 and 30 is 10.

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Answer 2

Answer:

32=2×2×2×2×2

54=2×3×3×3×3

common factors in both number is 2

so the HCF of 32 and 54 is 2.