Ncert Solutions for class 10 subject Maths Chapter 1 REAL NUMBERSin pdf Best Free NCERT Solutions for class 1 to 12 in pdf NCERT Solutions, cbse board, Maths, ncert Solutions for Class 10 Maths, class 10 Maths ncert solutions, REAL NUMBERS, Class 10, ncert solutions chapter 1 REAL NUMBERS, class 10 Maths, class 10 Maths ncert solutions, Maths ncert solutions class 10, Ncert Solutions Class 10 Mathematics Chapter 1 REAL NUMBERS

That means if we ide number a by b and q is our quotient and r is remainder then value of remainder will always less then deviser b .

If we ide 22 by 5 we get 4 as quotient and 2 is remainder

So we can write it as 22= 5*4+ 2

And you can see that value of remainder is less then deviser

**Composite number ** Every composite number can be expressed (factorized) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.

**The Fundamental Theorem of Arithmetic** If we ignore the order, any number which is more than 1, is either a prime number or can be written as unique product of prime number .

Or

Every composite number can be written in term of product of unique set of prime numbers ignoring their

**For example **

26 is a composite number and it can be written as 2*13 according to fundamental theorem you cannot get another set of value which product is 26.

So if we ignore the order of factors, prime factorization of all natural number is always unique.

for any prime number p which is ides a^2 then it will ide a also.

**For example:** 5 ides 100 then it will ide root of 100( 10) also.

for any rational number x which is written as p/q and if q can be written in form of 2^n*p^m and value of m and n are positive integer or equal to 0 then decimal expansion of x will terminate.

**For example** take x = 7/50

And 50 = 2*5*5 = 2^1*5^2

And powers 1 and 2 both are positive integer So value of x will terminate

for any rational number x which is written as p/q and if q canâ€™t be written in form of 2^n*p^m and value of m and n are positive integer or equal to 0 then decimal expansion of x will non-terminating repeating (recurring).

**For example **

X = 7/15

Here denominator 15 = 3*5

And it cannot be written in form of 2^n*5^m

x =0.4666â€¦â€¦â€¦â€¦â€¦

So x will non-terminating repeating (recurring) .

**Question 1. Use Euclidâ€™s ision algorithm to find the HCF of : (i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255**

**Answer: (i) **

Here 225 > 135 we always ide greater number with smaller one.

Divide 225 by 135 we get 1 quotient and 90 as remainder so that

225= 135*1 + 90

Divide 135 by 90 we get 1 quotient and 45 as remainder so that

135= 90*1 + 45

Divide 90 by 45 we get 2 quotient and no remainder so we can write it as

90 = 2*45+ 0

As there are no remainder so deviser 45 is our HCF

**(ii) **

38220>196 we always ide greater number with smaller one.

Divide 38220 by 196 then we get quotient 195 and no remainder so we can write it as

38220 = 196 * 195 + 0

As there is no remainder so deviser 196 is our HCF

**(iii)**

867>255we always ide greater number with smaller one.

ide 867 by 255 then we get quotient 3 and remainder is 102

so we can write it as

867 = 255 * 3 + 102

Divide 255 by 102 then we get quotient 2 and remainder is 51

So we can write it as

255 = 102 * 2 + 51

Divide 102 by 51 we get quotient 2 and no remainder

So we can write it as

102 = 51 * 2+ 0

As there is no remainder so deviser 51 is our answer

Part-1

Part-2

Part-2

**Answer:**

Let take a as any positive integer and b = 6.

Then using Euclidâ€™s algorithm we geta = 6q + rhere r is remainder and value of q is more than or equal to 0 and r = 0, 1, 2, 3, 4, 5 because 0 <= r < band the value of b is 6

So total possible formswill6q+0 , 6q+1 , 6q+2,6q+3,6q+4,6q+5

6q+06 is isible by 2 so it is a even number

6q+16 is isible by 2 but 1 is not isible by 2 so it is a odd number

6q+26 is isible by 2 and 2 is also isible by 2 so it is a even number

6q+36 is isible by 2 but 3 is not isible by 2 so it is a odd number

6q+46 is isible by 2 and 4 is also isible by 2 it is a even number

6q+56 is isible by 2 but 5 is not isible by 2 so it is a odd number

**So odd numbers will in form of 6q + 1, or 6q + 3, or 6q + 5**

**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?**

**Answer:**

To get the maximum number column here we always find HCF and for minimum number we find LCM

So can use Euclidâ€™s algorithm to find the HCF.

Here 616> 32 so always ide greater number with smaller one

When we ide 616 by 32 we get quotient 19 and remainder 8

So we can write it as

616 = 32 x 19 + 8

Now ide 32 by 8 we get quotient 4 and no remainder

So we can write it as

32 = 8 x 4 + 0

As there are no remainder so our HCF will 8

**So that maximum number of columns in which they can march is 8. **

**4. Use Euclidâ€™s ision lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.**

**Answer: **

Let take a as any positive integer and b = 3.

Then using Euclidâ€™s algorithm we geta = 3q + r here r is remainder and value of q is more than or equal to 0 and r = 0, 1, 2 because 0 < r < b and the value of b is 3 So our possible values will 3q+0 , 3q+1 and 3q+2

Now find the square of values

Use the formula (a+b)² = a² + 2ab +b² to open the square bracket

(3q)² = 9q² if we ide by 3 we get no remainder

we can write it as 3*(3q²) so it is in form of 3m here m = 3q²

(3q+1)² = (3q)² + 2*3q*1 + 1²

=9q² + 6q +1 now ide by 3 we get 1 remainder

so we can write it as 3(3q² + 2q) +1 so we can write it in form of 3m+1 and value of m is 3q² + 2q here

(3q+2)² = (3q)² + 2*3q*2 + 2²

=9q² + 12q +4 now ide by 3 we get 1 remainder

so we can write it as 3(3q² + 4q +1) +1 so we can write it in form of 3m +1 and value of m will 3q² + 4q +1

Square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

**Question 5. Use Euclidâ€™s ision lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.**

**Answer: **

According to Euclidâ€™s Division Lemma

Let take a as any positive integer and b = 9.

Then using Euclidâ€™s algorithm we get a = 9q + r here r is remainder and value of q is more than or equal to 0 and r = 0, 1, 2, 3, 4, 5 , 6 , 7 , 8, because 0 â‰¤r < b and the value of b is 9

Sp possible forms will 9q, 9q+1, 9q+2,9q+3,9q+4,9q+5,9q+6,9q+7 and 9q+8

to get the cube of these values use the formula

(a+b)³ = a³ + 3a²b+ 3ab² + b³

In this formula value of a is always 9q

So plug the value we get

(9q+b)³ = 729q³ + 243q²b + 27qb² + b³

Now ide by 9 we get quotient = 81q³ + 27q²b + 3qb² and remainder is b³

So we have to consider the value of b³

b = 0 we get 9m+0 = 9m

b = 1 then 1³ = 1 so we get 9m +1

b = 2 then 2³ = 8 so we get 9m + 8

b = 3 then 3³ = 27 and it is isible by 9 so we get 9m

b = 4 then 4³ =64 ide by 9 we get 1 as remainder so we get 9m +1

b=5 then 5³=125 ide by 9 we get 8 as remainder so we get 9m+8

b=6 then 6³=216 ide by 9 no remainder there so we get 9m

b=7 then 7³ = 343 ide by 9 we get 1 as remainder so we get 9m+1

b=8 then 8³ = 512 ide by 9 we get 8 as remainder so we get 9m+8

So all values are in form of 9m , 9m+1 or 9m+8

**Question 1. Express each number as a product of its prime factors: (i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429**

**Answer:**

(i) 140 = 2 x 2 x 5 x 7 =2² x 5 x 7

(ii)156 =2 x 2 x 3 x 13 =2² x 3 x 13

(iii)3825 =3 x 3 x 5 x 5 x 17 =3² x 5² x 17

(iv)5005 =5 x 7 x 11 x 13 =5 x 7 x 11 x 13

(v)7429 =17 x 19 x 23 =17 x 19 x 23

**Question 2. Find the LCM and HCF of the following pairs of integers and verify that LCM Ã— HCF = product of the two numbers. 26 and 91 (ii) 510 and 92 (iii) 336 and 54**

**Answer:**

**(i)**

26 = 2 x 13

91 =7 x 13

HCF = 13

LCM =2 x 7 x 13 =182

Product of two value 26 x 91 = 2366

Product of HCF and LCM 13 x 182 = 2366

**Hence, product of two numbers = product of HCF Ã— LCM **

**(ii)**

510 = 2 x 3 x 5 x 17

92 =2 x 2 x 23

HCF =2

LCM =2 x 2 x3 x 5 x 17 x 23 = 23460

Product of both values 510x92 = 46920

Product of HCF and LCM 2x23460 =46920

**Hence, product of two numbers = product of HCF Ã— LCM **

**(iii)**

336 = 2 x 2 x 2 x 2 x 3 x 7

54 = 2 x 3 x 3 x 3

HCF = 2 x 3 = 6

LCM = 2 x 2 x 2 x 2 x 3 x 3 x 3 x 7 =3024

Product of number 336 x 54 =18144

Product of HCF and LCM 6 x 3024 = 18144

**Hence, product of two numbers = product of HCF Ã— LCM **

**Question 3. Find the LCM and HCF of the following integers by applying the prime factorization method. (i) 12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25**

**Answer:**

**(i) **

12 = 2 x 2 x 3

15 =3 x 5

21 =3 x 7

HCF = 3

LCM = 2 x 2 x 3 x 5 x 7 = 420

**(ii)**

17 = 1 x 17

23 = 1 x 23

29 = 1 x 29

HCF = 1

LCM = 1 x 17 x 19 x 23 = 11339

**(iii)**

8 =1 x 2 x 2 x 2

9 =1 x 3 x 3

25 =1 x 5 x 5

HCF =1

LCM = 1 x 2 x 2 x 2 x 3 x 3 x 5 x 5 = 1800

**Question 4. Given that HCF (306, 657) = 9, find LCM (306, 657).**

**Answer:**

We have the formula that

Product of LCM and HCF = product of number

LCM x 9 = 306 x 657

Divide both side by 9 we get

LCM = (306 x 657)/ 9 = 22338

**Question 5. Check whether 6n can end with the digit 0 for any natural number n.**

**Answer:**

If any digit has last digit 10 that means It is isible by 10 And the factors of

10 = 2x5 So value 6n should be isible by 2 and 5 both 6n is isible by 2 but not isible by 5 So it can not end with 0.

**Answer:**

7 Ã— 11 Ã— 13 + 13

Take 13 common there we get

13 (7 x 11 +1 )

13(77 + 1 )

13 (78)

It is product of two numbers and both numbers are more than 1 so it is a composite number

7 Ã— 6 Ã— 5 Ã— 4 Ã— 3 Ã— 2 Ã— 1 + 5

Take 5 common there we get

5(7 x 6 x 4 x 3 x 2 x 1 +1)

5(1008 +1)

5(1009)

It is product of two numbers and both numbers are more than 1 so it is a composite number

**Question 7. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?**

**Answer: **

They will be meet again after LCM of both values at the starting point.

To get the LCM we have to factorize the number.

18 = 2 Ã— 3 Ã— 3

12 = 2 Ã— 2 Ã— 3

LCM = 2 Ã— 2 Ã— 3 Ã— 3 = 36

Therefore, they will meet together at the starting point after 36 minutes.

**Question 1. Prove that âˆš5 is irrational.**

**Answer:**

Let take âˆš5 as rational number

If a and b are two co prime number and b is not equal to 0.

We can write âˆš5 = a/b

Multiply by b both side we get

bâˆš5 = a

To remove root, Squaring on both sides, we get

5b² = a² â€¦â€¦â€¦â€¦â€¦(1)

Therefore, 5 ides a² and according to theorem of rational number, for any prime number p which is ides a² then it will ide a also.

That means 5 will ide a. So we can write

a = 5c

and plug the value of a in equation (1) we get

5b² = (5c)²

5b² = 25c²

Divide by 25 we get

b²/5 = c²

again using same theorem we get that b will ide by 5

and we have already get that a is ide by 5

but a and b are co prime number. so it is contradicting .

Hence âˆš5 is a non rational number

**Question 2. Prove that 3 + 2âˆš5 is irrational.**

**Answer:**

Let take that 3 + 2âˆš5 is a rational number.

So we can write this number as

3 + 2âˆš5 = a/b

Here a and b are two co prime number and b is not equal to 0

Subtract 3 both sides we get

2âˆš5 = a/b â€“ 3

2âˆš5 = (a-3b)/b

Now ide by 2 we get

âˆš5 = (a-3b)/2b

Here a and b are integer so (a-3b)/2b is a rational number so âˆš5 should be a rational number But âˆš5 is a irrational number so it contradict the fact

Hence result is 3 + 2âˆš5 is a irrational number

**Question 3. Prove that the following are irrationals: (i) 1/âˆš2 (ii) 7âˆš5 (iii) 6 + âˆš2 **

**Answer:**

**(i) **Let take that 1/âˆš2 is a rational number.

So we can write this number as

1/âˆš2 = a/b

Here a and b are two co prime number and b is not equal to 0

Multiply by âˆš2 both sides we get

1 = (aâˆš2)/b

Now multiply by b

b = aâˆš2

ide by a we get

b/a = âˆš2

Here a and b are integer so b/a is a rational number so âˆš2 should be a rational number But âˆš2 is a irrational number so it is contradict

Hence result is 1/âˆš2 is a irrational number

**(ii)** Let take that 7âˆš5 is a rational number.

So we can write this number as

7âˆš5 = a/b

Here a and b are two co prime number and b is not equal to 0

Divide by 7 we get

âˆš5) =a/(7b)

Here a and b are integer so a/7b is a rational number so âˆš5 should be a rational number but âˆš5 is a irrational number so it is contradict

Hence result is 7âˆš5 is a irrational number.

**(iii)** Let take that 6 + âˆš2 is a rational number.

So we can write this number as

6 + âˆš2 = a/b

Here a and b are two co prime number and b is not equal to 0

Subtract 6 both side we get

âˆš2 = a/b â€“ 6

âˆš2 = (a-6b)/b

Here a and b are integer so (a-6b)/b is a rational number so âˆš2 should be a rational number But âˆš2 is a irrational number so it is contradict

Hence result is 6 + âˆš2 is a irrational number

**Question 1. Without actually performing the long ision, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion: (i)13/3125 (ii)17/8 (iii)64/455 (iv)15/1600 (v)29/343 (vi)23/2³*5² (vii)129/2²* 5**

**Answer:**

(i) 13/3125

Factorize the denominator we get

3125 =5 x 5 x 5 x 5 x 5 = 5^{5}

So denominator is in form of 5^{m} so 13/3125 is terminating .

(ii) 17/8

Factorize the denominator we get

8 =2 x 2 x 2 = 2³

So denominator is in form of 2^{n} so 17/8 is terminating .

(iii) 64/455

Factorize the denominator we get

455 =5 x 7 x 13

There are 7 and 13 also in denominator so denominator is not in form of 2^{n}*5^{m} . hence 64/455 is not terminating.

(iv) 15/1600

Factorize the denominator we get

1600 =2 x 2 x 2 x2 x 2 x 2 x 5 x 5 = 2^{6} x 5²

so denominator is in form of 2^{n} x 5^{m}

Hence 15/1600 is terminating.

(v) 29/343

Factorize the denominator we get

343 = 7 x 7 x 7 = 7³

There are 7 also in denominator so denominator is not in form of 2^{n} x 5^{m}

Hence it is none - terminating.

(vi) 23/(2³ x 5²)

Denominator is in form of 2^{n} x 5^{m}

Hence it is terminating.

(vii) 129/(2² x 5^{7} x 7^{5} )

Denominator has 7 in denominator so denominator is not in form of 2^{n} x 5^{n}

Hence it is none terminating.

(viii) 6/15

ide nominator and denominator both by 3 we get 2/5

Denominator is in form of 5^{m} so it is terminating.

(ix) 35/50 ide denominator and nominator both by 5 we get 7/10

Factorize the denominator we get

10=2 x 5

So denominator is in form of 2^{n} x5^{m} so it is terminating

(x) 77/210.

simplify it by iding nominator and denominator both by 7 we get 11/30

Factorize the denominator we get

30=2 x 3 x 5

Denominator has 3 also in denominator so denominator is not in form of 2^{n} x 5^{n}

Hence it is none terminating.

**Question 2. Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.**

(i)13/3125 = 0.009375

(ii)17/8

(iii)64/455 none- terminating

(iv)15/1600

(v)29/343 it is none â€“ terminating

(vi)23/2^{3}*5^{2} = 23/200

(vii)129/2^{2}* 5^{7}*7^{5} it is none terminating

(viii) 6/15 = 2/5 = 0.4

(ix)35/50

(x)77/210 it is none terminating

**Question 3. The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form p , q you say about the prime factors of q?**

**Answer:**

(i) 43.123456789

it has certain number of digits so they can be represented in form of p/q . Hence they are rational number

As they have certain number of digit and the number which has certain number of digits is always terminating number and for terminating number denominator has prime factor 2 and 5 only .

(ii) 0.120120012000120000. . .

In this problem repetitions number are not same so it is not a irrational number

so prime factor of denominator Q will has a value which is not equal to 2 or 5. And irrational number is always none terminating

(iii) 43.__123456789__

In this number 0.123456789 repeating again and again so it is a rational number and it is none terminating so that the prime factor has a value which is not equal to 2 or 5

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