1. Use Euclid's division algorithm to find the HCF of:
1) 135 and 225
2) 196 and 38220
3) 867and 255
Answers
1) 135 and 225:-
Given numbers: 135 and 225
Here, 225>135.
So, we will divide greater number by smaller number.
Divide 225 by 135.
The quotient is 1 and remainder is 90.
225=135×1+90
Divide 135 by 90
The quotient is 1 and remainder is 45.
135=90×1+45
Divide 90 by 45.
The quotient is 2 and remainder is 0.
90=2×45+0
Thus, the HCF is 45.
2) 196 and 38220:-
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.
3) 867 and 225 :-
Step 1: First find which integer is larger.
867>255
Step 2: Then apply the Euclid's division algorithm to 867 and 255 to obtain
867=255×3+102
Repeat the above step until you will get remainder as zero.
Step 3: Now consider the divisor 225 and the remainder 102, and apply the division lemma to get
255=102×2+51
102=51×2=0
Since the remainder is zero, we cannot proceed further.
Step 4: Hence the divisor at the last process is 51.
So, the H.C.F. of 867 and 255 is 51.
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Step-by-step explanation:
By Euclid's division lemma,
225=135×1+90
r=90
135=90×1+45
r=4
90=45×2+0
So, H.C.F of 135 and 225 is 45
(ii) By Euclid's division lemma,
38220=196×195+0r=0
So, H.C.F of 38220 and 196 is 196
(iii) By Euclid's division lemma,
867=255×3+102
r=10
255=102×2+51
r=51
102=51×2+0
So, H.C.F of 867 and 255 is 51
The highest HCF among the three is 196.