Use Euclid's division algorithm to find the HCF of

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

(iii) 867 and 225

**Soln. :** (i) HCF of 135 and 225;

Applying the Euclid's lemma to 225 and 135, (where 225 > 135), we get

225 = (135 × 1) + 90, since 90 ≠ 0, therefore, applying the Euclid's lemma to 135 and 90, we have 135 = (90 × 1) + 45

But 45 ≠ 0

∴ Applying Euclid's lemma to 90 and 45, we get 90 = (45 × 2) + 0

Here, *r* = 0, so our procedure stops. Since, the divisor at the last step is 45,

∴ HCF of 225 and 135 is 45.

(ii) HCF of 196 and 38220 :

We start dividing the larger number 38220 by 196, we get 38220 = (196 × 195) + 0

Here, *r* = 0

∴ HCF of 38220 and 196 is 196.

(iii) HCF of 867 and 255

Here, 867 > 255

∴ Applying Euclid's Lemma to 867 and 255, we get

867 = (255 × 3) + 102, 102 ≠ 0

∴ Applying Euclid's Lemma to 255 and 102, we get

255 = (102 × 2) + 51, 51 ≠ 0

∴ Applying Euclid's Lemma to 102 and 51, we get

102 = (51× 2) + 0, *r* = 0

∴ HCF of 867 and 225 is 51.

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

**Soln. :** Let us consider a positive odd integer as ′*a*′.

On dividing ′*a*′ by 6, let *q* be the quotient and ′*r*′ be the remainder.

∴ Using Euclid's lemma, we have : *a* = 6*q* + *r* where 0 ≤ *r* < 6 *i.e*., *r* = 0, 1, 2, 3, 4 or 5 *i.e*.,
*a* = 6*q* + 0 = 6*q* or *a* = 6*q* + 1

or *a* = 6*q* + 2 or *a* = 6*q* + 3

or *a* = 6*q* + 4 or *a* = 6*q* + 5

But, *a* = 6*q*, *a* = 6*q* + 2, *a* = 6*q* + 4 are even values of ′*a*′.

[∵ 6*q* = 2(3*q*) = 2*m*_{1}, 6*q* + 2 = 2(3*q* + 1) = 2*m*_{2},

6*q* + 4 = 2(3*q* + 2) = 2*m*_{3}]

But ′*a*′ being an odd integer, we have :
*a* = 6*q* + 1, or *a* = 6*q* + 3, or *a* = 6*q* + 5

An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups have to march in the same number of columns. What is the maximum number of columns in which they can march?

SOLUTION:
**Soln. :** Total number of members = 616

∴ The total number of members are to march behind an army band of 32 members is HCF of 616 and 32.
*i.e*., HCF of 616 and 32 is equal to the maximum number of columns such that the two groups can march in the same number of columns.

∴ Applying Euclid's lemma to 616 and 32, we get

616 = (32 × 19) + 8 ∵ 8 ≠ 0

∴ Again, applying Euclid's lemma to 32 and 8, we get

32 = (8 × 4) + 0, *r* = 0

∴ HCF of 616 and 32 is 8

Hence, the required number of maximum columns = 8.

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

[Hint : Let *x* be any positive integer then it is of the form 3*q*, 3*q* + 1 or 3*q* + 2. Now square each of these and show that they can be rewritten in the form 3*m* or 3*m* +1.]

**Soln. :** Let us consider an arbitrary positive integer as '*x*' such that it is of the form

3*q*, (3*q* + 1) or (3*q* + 2).

For *x* = 3*q*, we have *x*^{2} = (3*q*)^{2}

⇒ *x*^{2} = 9*q*^{2} = 3(3*q*^{2}) = 3*m * ... (1)

Putting 3*q*^{2} = *m* where *m* is an integer.
*x* = 3*q* + 1,
*x*^{2} = (3*q* + 1)^{2} = 9*q*^{2} + 6*q* + 1

= 3(3*q*^{2} + 2*q*) + 1 = 3*m* + 1 ... (2)

Putting 3*q*^{2} + 2*q* = *m*, where *m* is an integer.

For *x* = 3*q* + 2,
*x*^{2} = (3*q* + 2)^{2}

= 9*q*^{2} + 12*q* + 4 = (9*q*^{2} + 12*q* + 3) + 1

= 3(3*q*^{2} + 4*q* + 1) + 1 = 3*m* + 1 ... (3)

Putting 3*q*^{2} + 4*q* +1 = *m*, where *m* is an integer.

From (1), (2) and (3)
*x*^{2} = 3*m* or 3*m* + 1

Thus, the square of any positive integer is either of the form 3*m* or 3*m* + 1 for some integer *m*.

Use Euclid's division lemma to show that cube of any positive integer is of the form 9*m*,

9*m* + 1 or 9*m* + 8.

**Soln. :** Let us consider an arbitrary positive integer *x* such that it is in the form of 3*q*, (3*q* + 1) or

(3*q* + 2).

For *x* = 3*q*
*x*^{3 }= (3*q*)^{3}= 27*q*^{3} = 9(3*q*^{3}) = 9*m* ... (1)

Putting 3*q*^{3} = *m*, where *m* is an integer.

For *x* = 3*q* + 1
*x*^{3 }= (3*q* + 1)^{3} = 27*q*^{3} + 27*q*^{2} + 9*q* + 1

= 9(3*q*^{3} + 3*q*^{2} + *q*) + 1 = 9*m* + 1 ... (2)

Putting (3*q*^{3} + 3*q*^{2} + *q*) = *m*, where *m* is an integer.

For *x* = 3*q* + 2,
*x*^{3} = (3*q* + 2)^{3 }= 27*q*^{3} + 54*q*^{2} + 36*q* + 8

= 9(3*q*^{3} + 6*q*^{2} + 4*q*) + 8 = 9*m* + 8 ... (3)

Putting (3*q*^{3} + 6*q*^{2} + 4*q*) = *m*, where *m* is an integer.

From (1), (2), (3) we have:
*x*^{3} = 9*m*, (9*m* + 1) or (9*m* + 8)

Thus, cube of any positive integer can be in the form 9*m*, (9*m* + 1) or (9*m* + 8) for some integer *m*.

Express each number as a product of its prime factors:

(i) 140

(ii) 156

(iii) 3825

(iv) 5005

(v) 7429

**Soln. :** (i) Using factor tree method, we have:

∴ 140 = 2 × 2 × 5 × 7 = 2^{2} × 5 × 7

(ii) Using factor tree method, we have :

∴ 156 = 2 × 2 × 3 × 13 = 2^{2} × 3 × 13

(iii) Using factor tree method, we have:

∴ 3825 = 3 × 3 × 5 × 5 × 17 = 3^{2} × 5^{2} × 17

(iv) Using factor tree method, we have:

∴ 5005 = 5 × 7 × 11 × 13

(v) Using the factor tree method, we have:

∴ 7429 = 17 × 19 × 23

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.

(i) 26 and 91

(ii) 510 and 92

(iii) 336 and 54

**Soln. :** (i) Using factor-tree method, we have:

⇒ 26 = 2 × 13 and 91 = 7 × 13

∴ LCM of 26 and 91 = 2 × 7 × 13 = 182

HCF of 26 and 91 = 13

Now, LCM × HCF = 182 × 13 = 2366 and

26 × 91 = 2366
*i.e*., LCM × HCF = Product of two numbers.

(ii) Using the factor tree method, we have :

⇒ 510 = 2 × 3 × 5 × 17 and 92 = 2 × 2 × 23

∴ LCM of 510 and 92 = 2 × 2 × 3 × 5 × 17 × 23

= 23460

HCF of 510 and 92 = 2

⇒ LCM × HCF = 23460 × 2 = 46920

510 × 92 = 46920
*i.e*., LCM × HCF = Product of two numbers.

(iii) Using the factor tree method, we have:

∴ 336 = 2 × 2 × 2 × 2 × 3 × 7

∴ 54 = 2 × 3 × 3 × 3

LCM of 336 and 54 = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 7

= 3024

HCF of 336 and 54 = 2 × 3 = 6

LCM × HCF = 3024 × 6 = 18144

Also 336 × 54 = 18144

Thus, LCM × HCF = Product of two numbers.

Find the LCM and HCF of the following integers by applying the prime factorisation method.

(i) 12, 15 and 21

(ii) 17, 23 and 29

(iii) 8, 9 and 25

**Soln. :** (i) we have

⇒ 12 = 2 × 2 × 3, 15 = 3 × 5 and 21 = 3 × 7

∴ HCF of 12, 15 and 21 is 3

LCM of 12, 15 and 21 is 2 × 2 × 3 × 5 × 7
*i.e*., 420

(ii) we have :

LCM of 17, 23 and 29 = 17 × 23 × 29 = 11339

(iii) we have :

⇒ 8 = 2 × 2 × 2 and 9 = 3 × 3 and 25 = 5 × 5

∴ HCF of 8, 9 and 25 is 1

LCM of 8, 9 and 25 is

2 × 2 × 2 × 3 × 3 × 5 × 5 = 1800

Given that HCF (306, 657) = 9,

find LCM (306, 657).

**Soln. :** Here, HCF of 306 and 657 = 9 we have :

LCM × HCF = Product of the numbers.

∴ LCM × 9 = 306× 657

⇒ LCM = = 34 × 657 = 22338

Thus, LCM of 306 and 657 is 22338.

Check whether 6^{n} can end with the digit 0 for any natural number *n*.

**Soln. :** Here, *n* is a natural number and let 6^{n} end

in 0.

∴ 6^{n} is divisible by 5.

But the prime factors of 6 are 2 and 3.

⇒ 6^{n} = (2 × 3)^{n}
*i.e*., In the prime factorisation of 6^{n}, there is no factor 5.

So, by the fundamental theorem of Arithmetic, every composite number can be expressed as a product of primes and this factorisation is unique apart from the order in which the prime factorisation occurs.

∴ Our assumption that 6^{n} ends in 0, is wrong.

Thus, there does not exist any natural number *n* for which 6^{n} ends with zero.

Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.

SOLUTION:
**Soln. :** We have 7 × 11 × 13 + 13 = 13[(7 × 11) + 1]

= 13[78]
*i.e*., 13 × 78 cannot be a prime number because it has a factors 13 and 78.

∴ 13 × 78 is a composite number.

Also 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 = 5[7 × 6 × 4 × 3 × 2 × 1 + 1] which is also not a prime number.

[∵ It has a factor 5]

Thus, '7 × 11 × 13 + 13' and '7 × 6 × 5 × 4 × 3 × 2 × 1 + 5' are composite numbers.

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?

SOLUTION:
**Soln. :** Time taken by Sonia to drive one round of the field = 18 minutes

Time taken by Ravi to drive one round of the field = 12 minutes

The L.C.M. of 18 and 12 gives the exact number of minutes after which they meet at the starting point again.

We have

⇒ 18 = 2 × 3 × 3 and 12 = 2 × 2 × 3

∴ LCM of 18 and 12 = 2 × 2 × 3 × 3 = 36

Thus, they will meet again at the starting point after 36 minutes.

Prove that is irrational.

SOLUTION:
**Soln. :** Let be a rational number.

∴ We can find integers *a* and *b* (where, *b* ≠ 0 and *a* and *b* are co-prime) such that

⇒ ...(1)

Squaring both sides, we have *a*^{2} = 5*b*^{2}

∴ 5 divides *a*^{2} ⇒ 5 divides *a* ...(2)

∴ *a* = 5*c*, where *c* is an integer.

∴ Putting *a* = 5*c* in (1), we have

⇒ 25*c*^{2} = 5*b*^{2} ⇒ 5*c*^{2} = *b*^{2}

⇒ 5 divides *b*^{2} ⇒ 5 divides *b* ...(3)

From (2) and (3) *a* and *b* have at least 5 as a common factor. *i.e*., *a* and *b* are not co-prime.

∴ Our supposition that is rational is wrong.

Hence, is irrational.

Prove that is irrational.

SOLUTION:
**Soln. :** Let is rational.

∴ We can find two co-prime integers ′*a*′ and '*b*' such that

... (1)

∵ *a* and *b* are integers,

⇒ from (1), is a rational

But this contradicts the fact that is irrational.

∴ Our supposition is wrong.

Hence, is an irrational.

Prove that the following are irrationals.

(i)

(ii)

(iii)

**Soln. :** (i) We have

Let be rational, and

∴ is rational

Let such that *a* and *b* are co-prime integers and *b* ≠ 0.

∴ ... (1)

Since, the division of two integers is rational.

∴ is a rational.

From (1), is a rational number which contradicts the fact that is irrational.

∴ Our assumption is wrong.

Thus, is irrational.

(ii) Let us suppose that is rational.

Let there be two co-prime integers *a* and *b*.

such that where *b* ≠ 0

Now,

⇒ is a rational

This contradicts the fact that is irrational.

∴ We conclude that is irrational.

(iii) Let us suppose that is rational.

∴ We can find two coprime integers *a* and *b* (*b* ≠ 0), such that

... (1)

= Rational [ ∵ *a* and *b* are integers,]

is a rational number.

From (1), is a rational number, which contradicts the fact that is an irrational number.

∴ Our supposition is wrong.

is an irrational number.

Without actually performing the long divison, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(x)

**Soln. :** (i)

∵ 3125 = 5 × 5 × 5 × 5 × 5 = 5^{5}, which is of the form (2^{m }× 5^{n})

∴ is a terminating decimal.
*i.e*., will have a terminating decimal expansion.

(ii)

∵ 8 = 2 × 2 × 2 = 2^{3} = 1 × 2^{3}

= 5^{0} × 2^{3}, which is of the form (2^{m} × 5^{n})

∴ will have a terminating decimal expansion.

(iii)

∵ 455 = 5 × 7 × 13, which is not of the form

2^{m} × 5^{n}

∴ will have a non-terminating repeating decimal expansion.

(iv)

∵ 1600 = 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5

= 2^{6} × 5^{2}, which is of the form 2^{m} × 5^{n}]

∴ will have a terminating decimal expansion.

(v)

∵ 343 = 7 × 7 × 7 = 7^{3}, which is not of the form 2^{m} × 5^{n}

∴ will have a non-terminating repeating decimal expansion.

(vi)

*i.e*., *q* = 2^{3} × 5^{2}, which is of the form 2^{m} × 5^{n}.

∴ will have a terminating decimal expansion.

(vii)
*i.e*., *q* = 2^{2} . 5^{7} . 7^{5}, which is not of the form 2^{m} × 5^{n}.

∴ will have a non-terminating repeating decimal expansion.

(viii)

which is of the form 2^{m} × 5^{n}.

∴ will have a terminating decimal expansion.

(ix)

∵ 50 = 2 × 5 × 5 = 2^{1} × 5^{2}, which is of the form 2^{m }× 5^{n}.

∴ will have a terminating decimal expansion.

(x)

∵ 210 = 2 × 3 × 5 × 7 = 2^{1}.3^{1}.5^{1}.7^{1}, which is not of the form 2^{m }× 5^{n}.

∴ will have a non-terminating repeating decimal expansion.

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

SOLUTION:
**Soln. :** (i)

Multiplying and dividing both sides by 2^{5}, we have

(ii)

Multiplying and dividing by 5^{3}, we have

(iii) , represents non-terminating repeating decimal expansion.

(iv)

Multiplying and dividing both sides, by 5^{4}, we have

(v) , represents non-terminating repeating decimal expansion.

(vi)

Multiplying and dividing by 5, we have

(vii) represents non-terminating repeating decimal expansion.

(viii)

Multiplying and dividing by 2, we have

(ix)

(x) , represents non-terminating repeating decimal expansion.

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 what can you say about the prime factors of *q* ?

(i) 43.123456789

(ii) 0.120120012000120000

(iii)

**Soln. :** (i) 43.123456789

∵ The given decimal expansion terminates.

∴ It is rational of the form

Hence, *p* = 43123456789 and *q* = 2^{9} × 5^{9}

Prime factors of *q* are 2^{9} and 5^{9}.

(ii) 0.120120012000120000 ...

∵ The given decimal expansion is neither terminating nor non-terminating repeating,

∴ It is irrational number, hence cannot be written in *p*/*q* form.

(iii)

∵ The given decimal expansion is non-terminating repeating.

∴ It is rational number.

... (1)

Multiplying both sides by 1000000000, we have

1000000000*x* = 43123456789.123456789 ... (2)

Subtracting (1) from (2), we have

(1000000000*x*) - *x* = (43123456789.123456789.......)

- 43.123456789

⇒ 999999999*x* = 43123456746

Here, *p* = 4791495194 and *q* = 111111111, which is not of the form 2^{m }× 5^{n} *i.e*., the prime factors of *q* are not of the form 2^{m} × 5^{n}.