# Chapter 1. Q3. Show that any positive integer is of the form 3q or, 3q + 1 or, 3q + 2 for some integer q.

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1 Q1. State whether the following are true or not: Chapter 1 (i) 3 l 93 (ii) 6 l 28 (iii) 0 l 4 (iv) 5 l 0 (v) -2 l 8 (vi) -7 l -35 (vii) 6 l 6 (viii) 8 l -8 (ix) 13 l -25 (x) 1 l -1 Q2. Show that every positive even integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some integer. Q3. Show that any positive integer is of the form 3q or, 3q + 1 or, 3q + 2 for some integer q. Q4. Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer. Q5. Show that n 2-1 is divisible by 8, if n is an odd positive integer. Q6. Prove that if x and y are odd positive integers, then x 2 + y 2 even but not divisible by 4. Q7. Prove that n 2 - n is divisible by 2 for every positive integer n. Q8. Shaw that the square o] any positive integer is of the farm 3m or, 3m + 1 for some integer m. Q9. Use Euclid's division Lemma to show that the cube of any positive integer is either of the form 9m, 9m + 1 or, 9m + 8 for some integer m. Q10. Show that one and only one out of n, n + 2 or, n + 4 is divisible by 3, where n is any positive integer. Q11. Prove that one of every three consecutive positive integers is divisible by 3. Q12. Use Euclid's division algorithm to find the HCF of 210 and 55. Q13. Use Euclid's division algorithm to find the HCF of 4052 and Q14. Find the HCF of 81 and 237 and express it as a linear combination of 81 and 237. Q15. Find the HCF of 65 and 117 and express it in the form 65 m n. Q16. If the HCF of 210 and 55 is expressible in the form 210 X y, find y. Q17. d is the HCF of 56 and 72,find x, y satisfying d = 56x + 72y. Also, show that x and y are not unique. Class X - Chapter 1 Page 1

2 Q18. A sweet seller has 420 Kaju burfis and 130 Badam burfis she wants to stack them in such a way that each stack has the same number, and they take up the least area of the tray. What is the number of burfis that can be placed in each stack for this purpose? Q19. Any 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? Q20. Find the largest number which divides 245 and 1029 leaving remainder 5 in each case. Q21. Find the largest number that divides 2053 and 967 and leaves a remainder of 5 and 7 respectively. Q22. Find the largest number that will divide 398, 436 and 542 leaving remainders 7,11 and 15 respectively. Q23. Two tankers contain 850 litres and 680 litres of petrol respectively. Find the maximum capacity of a container which can measure the petrol of either tanker in exact number of times. Q24. In a seminar, the number of participants in Hindi, English and Mathematics are 60, 84 and 108, respectively. Find the minimum number of rooms required if in each room the same number of participants are to be seated and all of them being in the same subject. Q25. Three sets of English, Hindi and Mathematics books have to be stacked in such a way that all the books are stored topic-wise and the height of each stack is the same. The number of English books is 96, the number of Hindi books is 240 and the number of Mathematics books is 336. Assuming that the books are of the same thickness, determine the number of stacks of English, Hindi and Mathematics books. Q26. Express each of the following positive integers as the product of its prime factors: (i) 140 (ii) 156 (iii) 234 Q27. Express each of the following positive integers as the product of its prime factors: (i) 3825 (ii) 5005 (iii) 7429 Q28. Determine the prime factorization of each of the following numbers: (i) (ii) Q29. Prove that there is no natural number for which 4 n ends with the digit zero. Q30. Show that there are infinitely many positive primes. Class X - Chapter 1 Page 2

3 Q31. Prove that every positive integer different from 1can be expressed as a product of a nonnegative power of 2 mid an odd number. Q32. Prove that a positive integer 11 is prime number, if no prime p less than or equal to n divides n. Q33. Find the HCF and LCM of 90 and 144 by the prime factorisation method. Q34. Find the HCF and LCM of 144, 180 and 192 by prime factorisation method. Q35. Find the HCF of 96 and 404 by prime factorisation method. Hence, find their LCM. Q36. Find the largest positive integer that will divide 398, 436 and 542 leaving remainders 7, 11 and 15 respectively. Q37. In a seminar, the number of participants in Hindi, English and Mathematics are 60,84 and 108 respectively. Find the minimum number of rooms required if in each room the same number of participants are to be seated and all of them being in the same subject. Q38. Three sets of English, Hindi and Mathematics books have to be Stacked in such a way that all the books are stored topic wise and the height of each stack is the same. The number of English books is 96, the number of Hindi books is 240 and the number of Mathematics books is 336. Assuming that the books are of the same thickness, determine the number of stacks of English, Hindi and mathematics books. Q39. There is a circular path around a sports field. Priya takes 18 minutes to drive one round of the field, while Ravish 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? Q40. In a school there are two sections - section A and section B of class X. There are 32 students in section A and 36 students in section B. Determine the minimum number of books required for their class library so that they can be distributed equally among students of section A or section B. Q41. Prove that 2 is an irrational number. Q42. Prove that 3 is an irrational number. Q43. Prove that 3 2 is irrational. Q44. Prove that 5 is an irrational number. Class X - Chapter 1 Page 3

4 Q45. Prove that 5-3 is an irrational number. Q46. Prove that is irrational Q47. Prove that is irrational. Q48. Show that there is no positive integer n for which n 1 + n + 1 is rational. Q49. Let a and b be positive integers. Show that 2 always lies between a b and a+2b a+b. Q50. Let a, b, c, d be positive rationals sucn that a + b = c + d, then either a = c and b = d or b and d are squares of rationals. Q51. Let a, b, c, p be rational numbers such that p is not a perfect cube. If a + bp 1/3 + c p 2/3 = 0, then prove that a = b = c = 0. Q52. For any positive real number x, prove that there exists an irrational number y such that 0 < y < x. Q53. Without actually performing the long division, state whether the following rational numbers will have terminating decimal expansion or a non-terminating repeating decimal expansion. Also, find the number of places of decimals after which the decimal expansion terminates. (i) 17 8 (ii) (iii) (iv) (v) (vi) Q54. What can you say about the prime factorisations of the denominators of the following rationals. (i) (ii) Class X - Chapter 1 Page 4

5 Class X - Chapter 1 Page 5

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