# NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Ex 1.1

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## NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Exercise 1.1

Question 1.
Euclid’s division algoritgm to find the HCF of:
(i) 135 and 225
(ii) 196 and 38220
(iii) 867 and 225
Solution:
(i) 135 and 225
Since 225 > 135, we apply the division leema to 225 and 135, to get
225 = 135 x 1 + 90
Since the remainder 90 ≠ 0, we can apply the division lemma to 135 and 90, to get 135 = 90 x 1 + 45
We consider the new divisor 90 and new remainder 45, and apply division lemmat to get 90 = 45 x 2 + 0
The remainder has now become zero, so the HCF of 135 and 225 is 45.

(ii) 196 and 38220
Since 38220 > 196, we apply the division leema to 38220 and 196, to get
38220 = 196 x 195 + 0
The remainder has now become zero, so the HCF of 38220 and 196 is 196

(iii) 867 and 255
Since 867 > 255, we apply the division lemma to 867 and 225, to get
867 = 255 x 3 + 102
Since the remainder 102 = 0, we apply the division lemma to 255 and 102, to get 255 = 102 x 2 + 51
We consider the new divisor 102 and new remainder 51, and apply division lemma to get 102 = 51 x 2 + 0
The remainder has now become zero, so the HCF of 867 and 255 is 51.

Question 2.
Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.
Solution:
Let a be any positive odd integer. We apply the division algorithm with a and b = 6.
Since 0 < r < 6, the positive remainders are 0,1, 2, 3, 4, 5.
Now putting the values r = 0,1, 5 we get
⇒ a = bq + r
⇒ a = 6q + 0 ⇒ a = 6q [r = 0]
⇒ a = 6q + 1 ⇒ a = 6q +1 [r = 1]
Similarly integers of the form 6q, 6q + 1,6q + 2, 6q + 3, 6q + 4, 6q + 5.

Question 3.
An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?
Solution:
The maximum number of columns is HCF of (616,32).
Now, let us apply Euclid’s division algorithm to find their HCF
616 = 32 x 16 + 8
Since the remainder 8 ≠ 0, we apply the division lemma, to get
32 = 8 x 4 + 0
So, the HCF of 616 and 32 is 8.
Therefore, the maximum number of columns is 8.

Question 4.
Use Euclid’s division lemma to show, that the square of any positive integer is either of the form 3m or 3m +1 for some integer m.
Solution:
[Hint : Let x be any positive integer then it is of the form 3q, 3q +1, or 3q + 2. Now square each of these and show that they can be written in the form 3m or 3m+ 1]

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