### Text Book Questions Solved

#### Exercise 1.1 (Page – 7)

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

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

i) Applying Euclid’s division algorithm on 135 and 225, to get

225 = 1 x 135 + 90

Again, applying Euclid’s division algorithm on 90 and 135, to get

135 = 1 x 90 + 45

Again, applying Euclid’s division algorithm on 45 and 90, to get

90 = 2 x 45 + 0

Now, HCF (225, 135) = HCF (133, 90) = HCF (90, 45) = 45.

ii) Applying Euclid’s division algorithm on 196 and 38220 to get

38220 = 195 x 196 + 0

∴ HCF (38220, 196) = 196

iii) Applying Euclid’s division algorithm on 867 and 225, to get

867 = 3 x 255 + 102

Again, applying Euclid’s division algorithm on 102 and 255, to get

255 = 2 x 102 + 51

Again, applying Euclid’s division algorithm on 51 and 102, to get

102 = 2 x 51 + 0

Now, HCF (867, 255) = HCF (255, 102) = HCF (102, 51) = 51.

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

Consider a, a positive integer, we apply division algorithm with q and b = 6, 0 r < 6 such that

a = 6q + r

As r < 6 . Therefore , possible remainders are 0, 1, 2, 3, 4, 5.

Number a can be 6q, 6q + 1 , 6q + 2, 6q + 3, 6q + 4 or 6q + 5

6q = 2(3q) = 2m;

6q + 1 = 2(3q) + 1 = 2m + 1;

6q + 2 = 2(3q + 1) = 2n;

6q + 3 = 2(3q + 1) + 1 = 2n + 1;

6q + 4 = 2(3q + 2) = 2t;

6q + 5 = 2(3q + 2) + 1 = 2t + 1.

We note 6q, 6q + 2 and 6q + 4 are of the form 2r, r ∈ N, which are even numbers; and 6q + 1, 6q + 3 and 6q + 5 are of the form 2r + 1, r ∈ N, which are odd numbers.

Hence, 6q + 1, 6q + 3 and 6q + 5 are positive and odd integers.

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?

To find maximum number of columns, we have to find HCF of 616 and 32.

Using Euclid’s division algorithm, we have

616 = 19 x 32 + 8

Now, applying  Euclid’s division algorithm on 32 and 8m we have

32 = 4 x 8 + 0

∴ HCF (616, 32) = HCF (32, 8) = 8

Hence, maximum number of columns is 8.

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

Answer : Consider positive integer x, with q and b = 3, 0 r < 3

x can be of the form 3q, 3q + 1 and 3q + 2 because, r = 0, 1 , 2

There exist three cases:

Case I : x = 3q

⇒ x2 = (3q)2 = 9q2 = 3m, m = 3q2  ………….. (i)

Case II : x = 3q + 1

⇒ x2 = (3q + 1)2 = 9q2 +6q + 1 = 3(3q2+ 2q)+ 1
= 3m + 1, m = 3q2+ 2q    ………….. (ii)

Case III : x = 3q + 2

⇒ x2 = (3q + 2)2 = 9q2 +12q + 4
= 3(3q2+ 4q + 1) + 1
= 3m + 1, m = 3q2+ 4q + 1   ………….. (iii)

Hence, from (i) , (ii) and (iii) , the square of any positive integer is either of the from 3m or 3m + 1.

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

Answer : Consider positive integer a, with q and b = 3, 0 r < 3.

can be of the form 3q, 3q + 1 and 3q + 2 because, r = 0, 1 , 2

Consider a = 3q

⇒ a3 = 27q3 = 9(3q3)= 9m, m = 3q3

Consider a = 3q + 1

⇒ a3 = (3q + 1)3 = 27q+ 27q2+ 9q + 1
= 9(3q+ 3q2+ q) + 1 = 9m + 1 ; where m = 3q+ 3q2+ q

And consider a = 3q + 2

⇒ a3 = (3q + 2)3 = 27q+ 54q2+ 36q + 8
= 9(3q+ 6q2+ 4q) + 8 = 9m + 8 ; where m = 3q+ 3q2+ q

Hence, cube of any positive integer is of the form 9m, 9m + 1 or 9m  + 8.

#### Exercise 1.2 (Page – 11)

Question 1 : Express each number as a product of its prime factors:

i) 140
ii) 156
iii) 3825
iv) 5005
v) 7429     Question 2 : Find the LCM and HCF of the following pairs of integers and verify that LCM x HCF = product of the two numbers.

i) 26 and 91
ii) 510 and 92
iii) 336 and 54     Question 3 : 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   Question 4 : Given that HCF (306, 657) = 9, find LCM (306, 657). Question 5 : Check whether 6n can end with the digit 0 for any natural number n. Question 6 : Explain why 7 x 11 x 13 + 13 and 7 x 6 x 5 x 4 x 3 x 2 x 1 + 5 are composite numbers. 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, ang go in the same direction. After how many minutes will they meet again at the starting point? #### Exercise 1.3 (Page -14)

Question 1 : Prove that √5 is irrational. Question 2 : Prove that 3 + 2√5 is rational.  Question 3 : Prove that the following are irrationals:

i) 1/√2
ii) 7√5
iii) 6 + √2    #### Exercise 1.4 (Page  – 17-18)

Question 1 : Without actually performing the long division, 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) 16/1600
v) 29/343
vi) 23/2352
vii) 129/225775
viii) 6/15
ix) 35/50
x) 77/210            Question 2 : Write down the decimal expansions of those rational numbers in Question 1 above which have a terminating decimal expansions. 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, what can you say about the prime factors of q?

i) 43.123456789
ii) 0.120120012000120000………    