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 repea...