Real Numbers

 1. Use Euclid’s division algorithm to find the HCF of:

i. 135 and 225

ii. 196 and 38220

iii. 867 and 255

Solutions:

i. 135 and 225

Since 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, in the last step, the divisor is 45, therefore, HCF (225,135) = 45.

ii. 196 and 38220

Since 38220 is greater than 196. Therefore, by Euclid’s division algorithm, we have,

38220 = 196 × 195 + 0

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

iii. 867 and 255

Since 867 is greater than 255. Therefore, by Euclid’s division algorithm, we have,

867 = 255 × 3 + 102

Now remainder 102 ≠ 0, therefore taking 255 as divisor and applying the division lemma method, we get,

255 = 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, in the last step and the divisor is 51, therefore, HCF (867,255) = 51