Use Euclid's division algorithm to find the HCF of: (i) 135and225 (ii) 196and38220 (iii) 867and255
Answers
Answer:
(i)HCF=45
(ii)HCF=195
(iii)HCF=51
Step-by-step explanation:
Given :
- 135 and 225
- 196 and 38220
- 867 and 255
To find :
- HCF by Euclid's Division Algorithm =?
Step-by-step explanation :
(i). 135 and 225
Euclid's division lemma :
Let a and b be any two positive Integers .
Then there exist two unique whole numbers q and r such that
a = bq + r ,
0 ≤ r <b
Now ,
Clearly, 225 > 135
Start with a larger integer , that is 225.
Applying the Euclid's division lemma to 225 and 135, we get
225 = 135 x 1 + 90
Since the remainder 90 ≠ 0, we apply the Euclid's division lemma to divisor 135 and remainder 90 to get
135 = 90 x 1 + 45
We consider the new divisor 90 and remainder 45 and apply the division lemma to get
90 = 45 x 2 + 0
Now, the remainder at this stage is 0.
So, the divisor at this stage, ie, 45 is the HCF of 225 and 135.
(ii). 196 and 38220
Euclid's division lemma :
Let a and b be any two positive Integers .
Then there exist two unique whole numbers q and r such that
a = bq + r ,
0 ≤ r <b
Now ,
Clearly, 38220 > 196
Start with a larger integer , that is 38220.
Applying the Euclid's division lemma to 38220 and 196, we get
38220 = 196 x 195 + 0
Now, the remainder at this stage is 0.
So, the divisor at this stage, ie, 196 is the HCF of 38220 and 196.
(iii) . 867 and 255
Euclid's division lemma :
Let a and b be any two positive Integers .
Then there exist two unique whole numbers q and r such that
a = bq + r ,
0 ≤ r <b
Now ,
Clearly, 867 > 255
Start with a larger integer , that is 255.
Applying the Euclid's division lemma to 867 and 255, we get
867 = 255 x 3 + 102
Since the remainder 102 ≠ 0, we apply the Euclid's division lemma to divisor 255 and remainder 102 to get
255 = 102 x 2 + 51
We consider the new divisor 102 and remainder 51 and apply the division lemma to get
102 = 51 x 2 + 0
Now, the remainder at this stage is 0.
So, the divisor at this stage, ie, 51 is the HCF of 867 and 255.