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NCERT Solutions for Class 10Maths
Chapter 1 – Real Numbers
Exercise 1.1
1. Use Euclid’s division algorithm to find the HCF of:
(i) 135 and 225
Ans: We have to find the HCF of 135 and 225 by using Euclid’s division
algorithm.
According to Euclid’s division algorithm, the HCF of any two positive integers a
and b , where a b is found as :
● First find the values of q and r , where a bq r , 0 r b .
● If r 0 , the HCF is b . If r 0 , apply Euclid’s lemma to b and r .
● Continue steps till the remainder is zero. When we get the remainder zero,
divisor will be the HCF.
Let a 225 and b 135 .
Since, a b
Using division algorithm, we get
a bq r
225 135 1 90
Here,
b 135
q 1
r 90
Since r 0 , we apply the Euclid’s lemma to b (new divisor) and r (new
remainder). We get
135 90 1 45
Here,
b 90
q 1
r 45
Since r 0 , we apply the Euclid’s lemma to b and r . We get
90 2 45 0
Now, we get r 0 , thus we can stop at this stage.
Class X Maths www.vedantu.com 1
, When we get the remainder zero, divisor will be the HCF.
Therefore, the HCF of 135 and 225 is 45 .
(ii) 196 and 38220
Ans: We have to find the HCF of 196 and 38220 by using Euclid’s division
algorithm.
According to Euclid’s division algorithm, the HCF of any two positive integers a
and b , where a b is found as :
● First find the values of q and r , where a bq r , 0 r b .
● If r 0 , the HCF is b . If r 0 , apply Euclid’s lemma to b and r .
● Continue steps till the remainder is zero. When we get the remainder zero,
divisor will be the HCF.
Let a 38220 and b 196 .
Since, a b
Using division algorithm, we get
a bq r
38220 196 195 0
Here,
b 196
q 195
r 0
Since, we get r 0 , thus we can stop at this stage.
When we get the remainder zero, divisor will be the HCF.
Therefore, the HCF of 196 and 38220 is 196 .
(iii) 867 and 255
Ans: We have to find the HCF of 867 and 255 by using Euclid’s division
algorithm.
According to Euclid’s division algorithm, the HCF of any two positive integers a
and b , where a b is found as :
● First find the values of q and r , where a bq r , 0 r b .
● If r 0 , the HCF is b . If r 0 , apply Euclid’s lemma to b and r .
● Continue steps till the remainder is zero. When we get the remainder zero,
divisor will be the HCF.
Let a 867 and b 255 .
Since, a b
Class X Maths www.vedantu.com 2
, Using division algorithm, we get
a bq r
867 255 3 102
Here,
b 255
q3
r 102
Since r 0 , we apply the Euclid’s lemma to b (new divisor) and r (new
remainder). We get
255 102 2 51
Here,
b 102
q2
r 51
Since r 0 , we apply the Euclid’s lemma to b and r . We get
102 51 2 0
Now, we get r 0 , thus we can stop at this stage.
When we get the remainder zero, divisor will be the HCF.
Therefore, the HCF of 867 and 255 is 51 .
2. Show that any positive odd integer is of the form, 6q 1 or 6q 3 , or 6q 5
, where q is some integer.
Ans: Let a be any positive integer and b 6 .
Then, by Euclid’s division algorithm we get a bq r , where, 0 r b .
Here, 0 r 6 .
Substitute the values, we get
a 6q r
If r 0 , we get
a 6q 0
a 6q
If r 1, we get
a 6q 1
If r 2 , we get
a 6q 2 and so on
Therefore, a 6q or 6q 1 or 6q 2 or 6q 3 or 6q 4 or 6q 5 .
Class X Maths www.vedantu.com 3
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