Exercise 1.1

Question 1. Use Euclid'e division algorithm to find the HCF of :-

(i). 135 and 225

Solution :-

By Euclid's division algorithm,we have

225 = 135×1+90

135 = 90×1+45

90 = 45×2+0

Hence,the HCF of 135 and 225 is 45.

(ii). 196 and 38220

Solution :-

By Euclid's division algorithm, we have

38220 = 196×195+0

Hence,the HCF of 196 and 38220 is 196.

(iii). 867 and 255

Solution :-

By Euclid's division algorithm,, we have

867 = 255×3+102

255 = 102×2+51

102 = 51×2+0

Hence, 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 a positive odd integer. Also,let 'q' be the quotient and 'r' the remainder. After dividing 'a' by 6.

Then, a = 6q+1 ,where 'r' is greater than 0 and 6 is greater than 'r'.

Putting r = 0,1,2,3,4 and 5,we get

a = 6q

a = 6q+1

a = 6q+2

a = 6q+3

a = 6q+4

a = 6q+5

But, a = 6q ,a = 6q+2 ,a = 6q+4 are even.

Hence,when 'a' is odd,it is of the form 6q+1, 6q+3 and 6q+5 for some integer 'q'.

Proved

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 :-

Maximum number of columns = HCF of 616 and 32.

For finding the HCF,we should apply Euclid's division algorithm.

On applying Euclid's division algorithm,we have

616 = 32×19+8

32 = 8×4+0

The remainder has now become zero,so we stop.

The HCF of 616 and 32 is 8.

Therefore,the maximum number of columns in which an army contingent of 616 members can march behind an army band of 32 members in a parade is 8.

Question 4.