((i) 135 and 225

Step 1: Since 225 > 135, apply Euclid's division lemma

let a =225 and b=135 to find q and r such that 225 = 135q+r, (0≤r<135)

On dividing 225 by 135 we get quotient as 1 and remainder as 90

Using Euclid's division lemma

i.e 225 = 135 × 1 + 90

Step 2: Remainder r which is 90 now we apply Euclid's division lemma to a = 135 and b = 90 to find whole numbers q and r such that

135 = 90 × q + r (0≤r<90)

On dividing 135 by 90 we get quotient as 1 and remainder as 45

i.e 135 = 90 × 1 + 45

Step 3: Again remainder r = 45 so we apply Euclid's division lemma to a = 90 and b = 45 to find q and r such that

90 = 45 × q + r (0≤r<45)

On dividing 90 by 45 we get quotient as 2 and remainder as 0

i.e 90 = 2×45 + 0

Step 4: Since the remainder is zero

Since the divisor at this stage is 45, therefore HCF of 135 and 225 is 45.

(ii) 196 and 38220

Step 1: Since 38220 > 196, apply Euclid's division lemma

let a =38220 and b=196 to find whole numbers q and r such that

Using Euclid's division lemma

38220 = 196 q + r, (0≤r<196)

On dividing 38220 we get quotient as 195 and remainder r as 0

i.e 38220 = 196 × 195 + 0

Since the remainder is zero therefore divisor at this stage is HCF of 196 and 38220 is 196.

(iii) 867 and 255

Step 1: Since 867 > 255, apply Euclid's division lemma

Let a =867 and b=255 to find q and r Using Euclid's division lemma

867 = 255q + r, (0≤r<255)

i.e 867 = 255 × 3 + 102

On dividing 867 by 255 we get quotient as 3 and remainder as 102

Step 2: Again applying Euclid's division lemma to a=255 and b= 102 to find whole numbers q and r such that

255 = 102q + r where (0≤r<102)

On dividing 255 by 102 we get quotient as 2 and remainder as 51

i.e 255 = 102 × 2 + 51

Step 3: Again remainder 51 is non zero, so we apply the division lemma to a=102 and b= 51 to find whole numbers q and r such that

102 = 51 q + r where (0≤r<51)

On dividing 102 by 51 quotient is 2 and remainder is 0

i.e 102 = 51 × 2 + 0

Since the remainder is zero therefore divisor at this stage is HCF of 867 and 255.

Therefore HCF is 51.