# Real NumbersExercise – 1.1

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

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

(i) 135 and 225

Solution:- 225>135

Let a = 225, b =135

From Euclid's division algorithm,

a = b ×q + r, 0 ≤ r < b

225 = 135×1 + 90 (r≠0, When r is not equal to zero, the theorem is used again.)

135 = 90×1 + 45 (r≠0)

90 = 45×2 + 0 (r=0, The solution is terminated when r is equal to zero and the value of b is HCF.)

HCF = 45

(ii) 196  and 38220

Solution:- 38220>196

Let, a = 38220, b=196

From Euclid's division algorithm,

a = b×q + r, 0≤ r < b

38220 = 196×195 + 0 (r=0, The solution is terminated when r is equal to zero and the value of b is HCF.)

HCF = 196

(iii) 867 and 225

Solution:- 867>225

Let a = 867, b= 225

From Euclid's division algorithm,

a = b×q + r, 0≤ r < b

867 = 225×3 + 102 (r≠0)

225 = 102×2 + 51 (r≠0)

102 = 51×2 + 0 (r=0, The solution is terminated when r is equal to zero and the value of b is HCF.)

HCF = 51

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

Solution:-

we have to show _ a = 6q+1, 6q+3 या 6q+5

Let a be any positive odd integer and b = 6, then by Euclid's division algorithm,

a = 6q+r, r= 0,1,2,3,4,5 (0≤ r <6, The divisor cannot be greater than nor equal to the remainder)

Put the value of 'r',

If r = 2, so a = 6q

Similarly, Putting r = 1,2,3,4,5 will give a = 6q+1, 6q+2, 6q+3, 6q+4, 6q+5 respectively.

Hence, a positive odd integer will be of the form 6q+1, 6q+3 or 6q+5.

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 will be equal to HCF(616, 32).

Since, 616 > 32

Then, from Euclid's algorithm,

616 = 32 × 19 + 8 ( r ≠ 0, So 32 as the new dividend and 8 as the new divisor )

32 = 8 × 4 + 0 (r =0, then the value of b is HCF)

b = 8

HCF(616, 32) = 8

So the maximum number of columns will be 8.

4 . Use Euclid’s division lemma to show that the square of any positive integer is either of form 3m or 3m + 1 for some integer m.

Solution:

Let ‘a’ be any positive integer and b = 3

Then, from Euclid's algorithm,

a = bq + r  for some integers q ≥ 0 and r = 0,1,2 where 0 r < 3

therefore, a = 3q, a = 3q+1, a = 3q+2

According to the question, on squaring all the equations on both sides,

a2= 3m or 3m + 1 or 3m + 1

5. Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m+1 or 9m+8

Solution:

Let ‘a’ be any positive integer and b = 3

Then, from Euclid's algorithm,

a = bq + r  for some integers q ≥ 0 and r = 0,1,2 where 0 r < 3

When r = 0,

a = 3q

Hence, the cube of a positive integer will be of the form 9m, 9m+1 or 9m+8.