**Class 10, CBSE Board Mathematics**

**Chapter 1, Real Numbers**

__NCERT Mathematics Solutions__

**Maths Textbook Exercise 1.1 Solutions**

(Page 7)

Q.1: Use Euclid’s division algorithm to find the HCF of:

(i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255

Solution:

(i) We have,

a = bq + r

Applying division lemma to 225 and 135 we obtain,

225 = 135 x 1 + 90

and 135 = 90 x 1 + 45

and 90 = 45 x 2 + 0

Therefore, HCF of 225, 135 = 45

(ii) We have,

a = bq + r

Applying division lemma to 196 and 38220 we obtain,

38220 = 196 x 195 + 0

Therefore, HCF of 196 and 38220 = 196

(iii) We have,

a = bq + r

Applying division lemma to 867 and 255 we obtain,

867 = 255 x 3 + 102

255 = 102 x 2 + 51

102 = 51 x 2 + 0

Therefore, HCF of 867 and 255 is 51.

Q.2: Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or6q + 5, where

*q*is some integer.Solution:

Let

*a*be any positive number and*a*= 6. Then, by Euclid’s algorithm,*a*= 6

*q*+

*r*(0 ≤

*r*< 6)

say,

*r*= 0, 1, 2, 3, 4, 5or,

*a*= 6*q*or 6*q*+ 1 or 6*q*+ 2 or 6*q +*3 or 6*q*+ 4 or 6*q*+ 5Also, 6

*q*+ 1 = 2 × 3*q*+ 1 = 2k_{1}+ 1, where*k*_{1}is a positive integer,Similarly, 6

*q*+ 3 = (6*q*+ 2) + 1 = 2 (3*q*+ 1) + 1 = 2k_{2}+ 1, where*k*_{2}is a positive integer,and, 6

*q*+ 5 = (6*q*+ 4) + 1 = 2 (3*q*+ 2) + 1 = 2k_{3}+ 1, where*k*_{3}is a positive integer.From these we observe that 6

*q*+ 1, 6*q*+ 3, 6*q*+ 5 are of the form 2*k*+ 1. So, these numbers are not divisible by 2 and hence, are odd positive integers.Q.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:

To find the maximum number of columns, we have to find HCF of 616 and 32.

Applying Euclid’s algorithm to find the HCF we get,

616 = 32 × 19 + 8

32 = 8 × 4 + 0

Or, the HCF (616, 32) = 8.

Therefore, maximum number of column is 8.

Q.4: Use Euclid’s division lemma to show that the square of any positive integer is either of form 3

*m*or 3*m*+ 1 for some integer*m*.Solution:

a = bq + r;

Let

*a*be any positive integer,*b*= 3 and*r*= 0, 1, 2 because 0 ≤*r*< 3Then

*a*= 3*q*+*r*for some integer*q*≥ 0Therefore,

*a*= 3*q*+ 0 or 3*q*+ 1 or 3*q*+ 2From the above we can say that the square of any positive integer is either of the form 3

*m*or 3*m*+ 1.Q.5: Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9

*m*, 9*m*+ 1 or 9*m +*8.[Taking hint from the above do it yourselves]

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