# Real Numbers Exercise – 1.1 NCRT Download PDF || UP Board

# __Real Numbers__

__Exercise – 1.1__

__Real Numbers__

__Exercise – 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 ****6****q****+1, 6****q****+3 ****or**** 6****q****+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 ****3****m or**** 3****m ****+ 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,

^{2}= 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.

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