Question

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

Answer 1

Step-by-step explanation:

i) Answer : We are asked to find the HCF of 135 and 225 using Euclid’s division.

We know that the steps involved in Euclid’s division are

(1) We divide the large number with a small number and take the remainder.

(2) Next we divide the divisor used in the previous division with the remainder we got in the previous division and take the remainder.

(3) We carry out step (2) until we get the remainder as 0.

(4) The divisor used when the remainder is 0 will be the HCF of the given two numbers.

Now, let us apply the first step that is let us divide the number 225 with 135 then we get

⇒225=(135×1)+90

Here we can see that the divisor used in the previous division is 135 and the remainder is 90

Now, let us apply the second step that is let us divide the number 135 with 90 then we get

⇒135=(90×1)+45

Here we can see that the divisor used in the previous division is 90 and the remainder is 45

Now, let us apply the second step that is let us divide the number 90 with 45 then we get

⇒90=(45×2)+0

Here we can see that we got the remainder in the previous division as 0 when the divisor is 45

Now, let us apply the fourth step that is the divisor used to get the remainder 0 is the HCF

Therefore we can conclude that the HCF of 135 and 225 is 45

∴HCF(135,225)=45

ii) Answer : According to the definition of Euclid's theorem,

a=b×q+r where 0≤r<b.

Now,

⇒196 and 38220

⇒38220>196 so we will divide 38220 by 196

⇒38220=196×195+0

so 196 will be HCF.

I hope it will help you and mark me as a brainlist

Answer 2

Step-by-step explanation:

135 and 225

As we 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.

196 and 38220

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