Question

Use Euclid's division algorithm to find the HCF of
(i) 135 and 225
(ii) 196 and 38220
(iii) 867 and 255

Answer 1

SOLUTION :

(i) Given : Two positive integers are 225 and 135.

Here, 225 > 135.

Let a = 225  and b = 135

225 = 135 × 1 +90

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

Here, remainder = 90  ≠ 0, so take new dividend as 135 and new divisor as 90.

Let a = 135 and b= 90

135 = 90× 1 + 45

Here, remainder = 45  ≠ 0, so take new dividend as 90 and new divisor as 45

Let a = 90 and b= 45

90 = 45 ×  2 + 0

Here, remainder is zero and divisor is 45.

Hence ,H.C.F. of 225 and 135 is 45.

(ii) Given : Two positive integers are 38220 and 196.

Here,  38220 > 196.

Let a = 38220  and b = 196

38220 = 196 × 195 + 0

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

Here, remainder is zero and divisor is 196..

Hence ,H.C.F. of 38220 and 196 is 196.

(iii) Given : Two positive integers are 867 and 255.  

Here,  867 > 225.

Let a = 867 and b= 225

867 = 225 × 3 + 192

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

Here, remainder = 192  ≠ 0, so take new dividend as 225 and new divisor as 192

Let a = 225 and b= 92

225 = 192 × 1 + 33

Here, remainder = 33  ≠ 0, so take new dividend as 192 and new divisor as 33.

Let a = 192 and b= 33

192 = 33 × 5 + 27

Here, remainder = 27  ≠ 0, so take new dividend as 33 and new divisor as 27

Let a = 33 and b= 27

33 = 27 × 1 + 6

Here, remainder = 6  ≠ 0, so take new dividend as 27 and new divisor as 6

Let a = 27 and b= 33

27 = 6 × 4 + 3

Here, remainder = 3  ≠ 0, so take new dividend as 6 and new divisor as 3.

Let a = 6 and b= 3

6 = 3 × 2  + 0

Here, remainder is zero and divisor is 3..

Hence ,H.C.F. of 867 and 255 is 3.

HOPE THIS ANSWER WILL HELP YOU…

Answer 2

i. 135 and 225

As you can see, from the question 225 is greater than 135. Therefore, by Euclid’s division algorithm, we have,

225 = 135 × 1 + 90

Now, remainder 90 ≠ 0, thus again using division lemma for 90, we get,

135 = 90 × 1 + 45

Again, 45 ≠ 0, repeating the above step for 45, we get,

90 = 45 × 2 + 0

The remainder is now zero, so our method stops here. Since, in the last step, the divisor is 45, therefore, HCF (225,135) = HCF (135, 90) = HCF (90, 45) = 45.

Hence, the HCF of 225 and 135 is 45.

ii. 196 and 38220

In this given question, 38220>196, therefore the by applying Euclid’s division algorithm and taking 38220 as divisor, we get,

38220 = 196 × 195 + 0

We have already got the remainder as 0 here. Therefore, HCF(196, 38220) = 196.

Hence, the HCF of 196 and 38220 is 196.

iii. 867 and 225

As we know, 867 is greater than 225. Let us apply now Euclid’s division algorithm on 867, to get,

867 = 225 × 3 + 102

Remainder 102 ≠ 0, therefore taking 225 as divisor and applying the division lemma method, we get,

225 = 102 × 2 + 51

Again, 51 ≠ 0. Now 102 is the new divisor, so repeating the same step we get,

102 = 51 × 2 + 0

The remainder is now zero, so our procedure stops here. Since, in the last step, the divisor is 51, therefore, HCF (867,225) = HCF(225,102) = HCF(102,51) = 51.

Hence, the HCF of 867 and 225 is 51.