Real Numbers

Class X

Chapter 1

NCERT Text Book Solution
Exercise 1.1

Q.No.1. Use Euclid division Algorithm to find out the HCF of the following numbers.

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

(i)

Solution:

According to Euclid division Algorithm, we know that

From question, it is clear that

a = 225

b = 135

Here, remainder is zero. Therefore, the step is over and divisor is 45.

Hence, HCF = 45 Ans.

(ii)

Solution:

According to Euclid division Algorithm, we know that

From question, it is clear that

a = 38220

b = 196

38220 = 196x195 + 0

Here, remainder is zero. Therefore, the step is over and divisor is 196.

Hence, HCF = 196 Ans.

(iii)

Solution:

According to Euclid division Algorithm, we know that

From question, it is clear that

a = 867

b = 255

867 = 255x3 + 102

255 = 102x2 + 51

102 = 51x2 + 0

Here, remainder is zero. Therefore, the step is over and divisor is 51.

Hence, HCF = 51 Ans.

Q.No. 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:

According to Euclid division Algorithm, we know that

Here,

b = 6

Since

The possible remainders are 0, 1, 2, 3, 4 and 5.

That is a can be

6q, or 6q + 1, or 6q + 2, or 6q + 3, or 6q + 4, or 6q + 5, where q is quotient. However, since a is odd, a cannot be 6q, or 6q + 2, or 6q + 4 (Since they are divisible by 2)

Therefore, any odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5.

Q.No. 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:

According to Euclid division Algorithm, we know that

Here,

a = 616

b = 32

616 = 32x19 + 8

32 = 8x4 + 0

Here, remainder is zero. Therefore, the step is over and divisor is 8.

Hence, HCF = 8.

Therefore, maximum number of column is 8 Ans.

Q.No. 4. Use Euclid division lemma to show that thee square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

Solution:

According to Euclid division Algorithm, we know that

Here,

a = x

b = 3

Since

The possible remainders are 0, 1, and 2.

That is x can be

3q, or 3q + 1, or 3q + 2 where q is quotient.

Now, squaring these

9q², or 9q² + 6q + 1, or 9q² + 12q + 4

When these are divided by 3

Form of these will be

3x3q², or 3(3q² + 2q) + 1, or 3(3q² + 4q + 1) + 1

Therefore, they can be written in the form

3m or 3m + 1.

Q.No. 5. Use Euclid division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.

Solution:

According to Euclid division Algorithm, we know that

Here,

a = x

b = 3

Since

The possible remainders are 0, 1, 2,3 , 4, 5, 6, 7, 8.

That is x can be

9q, or 9q + 1, or 9q + 2, 9q + 3, 9q + 4, 9q + 5, 9q + 6, 9q + 7 or 9q + 8 where q is quotient.

Now, cubing these

Therefore, cube of these can be written in the form

Cube of 9q = 729q³

Form = 9x81q³ = 9m

Cube of 9q + 1= 729q³ + 243q² + 27q + 1

Form = 9(81xq³ + 27q² + 3q) + 1 = 9m + 1

and so on .....................

9m or 9m + 1.