pairs of numbers are co-prime:

(i) 231, 396 (ii) 847, 2160

Solution : Let us find the HCF of each pair of numbers.

(i) 396 = 231 × 1 + 165

231 = 165 × 1 + 66

165 = 66 × 2 + 33

66 = 33 × 2 + 0

Therefore, HCF = 33. Hence, numbers are not co-prime.

(ii) 2160 = 847 × 2 + 466

847 = 466 × 1 + 381

466 = 381 × 1 + 85

381 = 85 × 4 + 41

85 = 41 × 2 + 3

41 = 3 × 13 + 2

3 = 2 × 1 + 1

2 = 1 × 2 + 0

Therefore, the HCF = 1. Hence, the numbers are co-prime.

Sample Question 2: Show that the square of an odd positive integer is of the form

8m + 1, for some whole number m.

Solution: Any positive odd integer is of the form 2q + 1,

where q is a whole number.

Therefore, (2q + 1)2 = 4q2 + 4q + 1

= 4q (q + 1) + 1,----- (1)

q (q + 1) is either 0 or even.

So, it is 2m, where m is a whole number.

Therefore, (2q + 1)2 = 4.2 m + 1 = 8 m + 1. [From (1)]

Sample Question 3: Prove that √2 + √3 is irrational.

Solution : Let us suppose that √2 + √3 is rational. Let 2 + √3 = a , where a is

rational.

Therefore, √2 = a− √3

Squaring on both sides, we get

2 = a

^{2}

**+ 3 – 2a√3**

2a√3 = a

^{2}

**+ 3 -2**

2a√3 = a

^{2}

**+ 1**

Therefore,

√3 =

**(a**

^{2}+ 1)/2a

Sample Question 4:

Show that the square of an odd positive integer can be of the form 6q + 1 or 6q + 3 for some integer q.

Solution :

Thus, an odd positive integer can be of the form 6m + 1, 6m + 3, or 6m + 5

Thus we have:

(6 m +1)2 = 36 m2 + 12 m + 1 = 6 (6 m2 + 2 m) + 1 = 6 q + 1, q is an integer

(6 m + 3)2 = 36 m2 + 36 m + 9 = 6 (6 m2 + 6 m + 1) + 3 = 6 q + 3, q is an integer

(6 m + 5)2 = 36 m2 + 60 m + 25 = 6 (6 m2 + 10 m + 4) + 1 = 6 q + 1, q is an integer.

Thus, the square of an odd positive integer can be of the form 6q + 1 or 6q + 3.

**Now Solve these Questions**

1. Show that the square of any positive integer is either of the form 4q or 4q + 1 for some integer q.

2. Show that cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3, for some integer m.

3. Show that the square of any positive integer cannot be of the form 5q + 2 or 5q + 3 for any integer q.

4. Show that the square of any positive integer cannot be of the form 6m + 2 or 6m + 5 for any integer m.

5. Show that the square of any odd integer is of the form 4q + 1, for some integer q.

6. If n is an odd integer, then show that n2 – 1 is divisible by 8.

7. Prove that if x and y are both odd positive integers, then x2 + y2 is even but not divisible by 4.

8. Use Euclid’s division algorithm to find the HCF of 441, 567, 693.

9. Using Euclid’s division algorithm, find the largest number that divides 1251, 9377 and 15628 leaving remainders 1, 2 and 3, respectively.

10. Prove that √ 3+ √ 5 is irrational.

11. Show that 12n cannot end with the digit 0 or 5 for any natural number n.

12. On a morning walk, three persons step off together and their steps measure 40 cm, 42 cm and 45 cm, respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?

13. Show that the cube of a positive integer of the form 6q + r, q is an integer and

r = 0, 1, 2, 3, 4, 5 is also of the form 6m + r.

14. Prove that one and only one out of n, n + 2 and n + 4 is divisible by 3, where n is

any positive integer.

15. Prove that one of any three consecutive positive integers must be divisible by 3.

16. For any positive integer n, prove that n3 – n is divisible by 6.

17. Show that one and only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible

by 5, where n is any positive integer.

[Hint: Any positive integer can be written in the form 5q, 5q+1, 5q+2, 5q+3,5q+4].

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