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

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

Solution:

Euclid’s division lemma states that for any two positive integers, say ‘a’ and ‘b’, the condition ‘a = b q +r‘, where 0 ≤ r < b always holds true
If, r = 0 then b, is called HCF of a and b.

(i) Here 225 > 135. Applying division lemma to 225 and 135, we get
225 = 135 x 1 + 90
Since the remainder 90 ≠ 0, again apply the division lemma to 90 and 135
135 = 90 x 1 + 45
Since now remainder is 45 ≠ 0, apply division algorithm to 45 and 90
90 = 45 x 2 + 0
The remainder now becomes zero and the divisor is 45 here.
Hence the HCF of 135 and 225 is 45

(ii) Since 38220 > 196, Applying division lemma to 38220 and 196, we get
38220 = 196 x 195 + 0
The remainder becomes zero and the divisor is 196.
Hence the HCF of 38220 and 196 is 196.

(iii) Here 867 > 255. Applying division lemma to 867 and 255, we get
867 = 255 x 3 + 102
Since the remainder 102 ≠ 0, again apply the division lemma to 102 and 255
255 = 102 x 2 + 51
Since now remainder is 51 ≠ 0, apply division algorithm to 102 and 51
102 = 51 x 2 + 0
The remainder now becomes zero and the divisor is 51 here.
Hence the HCF of 867 and 255 is 51

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