REAL NUMBERS EUCLIDS DIVISON LEMMA: - If a and b are two positive integers (a> b) we can find two integers q and r such that a = bxq+r, where 0≤r<b A number which has only two factors, namely 1 and itself is called a prime number. A number which has more than two factors is called a composite number FUNDAMENTAL THEOREM OF ARITHMETIC: Every composite number can be expressed as product of primes in a unique way, except for the order of those primes Numbers which divide a given number are called its factors Numbers obtained by multiplying a given number with other numbers are called multiples of the given number or Numbers which are divisible by the given number are called its multiples. The HCF of a group of numbers is the product of the common factors with the least powers, taken from their prime factorizations. The LCM of a group of numbers is the product of all the factors with the highest powers, taken from their prime factorizations The product of a pair of numbers is always equal to the product of their LCM and HCF Numbers which do not have any common factors other than 1 are called co-primes and their HCF is 1 and their LCM is their product The HCF of two consecutive numbers is 1 and the LCM is their product The HCF of two consecutive even numbers is 2 The HCE of two consecutive odd numbers is 1 and their LCM is their product The HCF of a group of numbers is always a factor of their LCM or LCM of a group of number is always a multiple of their HCF The unit’s digit of a number can be 0 if and only if the prime factorization of that number has both 2 and 5 If the prime factorization of a number has 5 but does not have 2 then the units digit of that number will be 5 If c is a factor of a and b, then c is a factor of a + b and a-b. If a prime number p is a factor of a2 then p is a factor of ‘a’. Every real number can be expressed as a decimal. If the real number is an irrational then the decimal will be non-terminating, non-repeating. If the real number is a rational then the decimal will either be terminating or nonterminating repeating. If p/q is a rational number in its simplest form and the prime factorization of q is in the form 2n x 5m then the decimal representation of it will be terminating. If p/q is a rational number in its simplestform and the prime factorization of q is not in the form of 2n×5m then the decimal representation of it will be non terminating repeating. If p/q is a rational number in its simplest form and if the prime factorization of q is in the form 2n×5mthen its decimal reparation will terminate after either n or m places whichever is greater. Non-terminating, non-repeating decimals represent irrational numbers. Terminating decimals and repeating decimals represent rational numbers.
PROBLEMS 1. Find the HCF of the following using Euclid's division algorithm a) 135 and 255 b) 867 and 255 c) 4052 and 12576 d) 196 and 38220 e) 441, 567 and 693 2. If the HCF of 408 and 1032 can be expressed in the form of 1032m-2040, find m. 3. If the HCF of 210 and 55 is equal to 1050 + 55y, find y. 4. Find the HCF of 81 and 237 and express it as 81a+ 237b 5. Find the HCF of 65 and 117 and express it in the form of 65m +117n 6. If d is the HCF of 56 and 72, find x and y satisfying d = 56x +72y 7. A sweet seller has 420 kajubarfis and 130 badambarfis. She wants to stack them in such a way that each stack has same number of barfis. What is the maximum number of barfis in each stack and what is the number of stacks for each type of barfis? 8. 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 same number of columns. What is the maximum number of columns in which they can March? 9. A merchant has 120 litres of oil of one kind, 180 litres of another kind and 240 litres of a third kind. He wants to sell the oil by filling the three kinds of oil in tins of equal capacity. What is the greatest capacity of such a tin? 10. Find the largest number that divides 2048 and 960 11. Find the largest number which divides 245 and 1029 leaving remainder 5 in each case 12. Find the greatest number which divides 285 and 1249 leaving remainders 9 and 7 respectively 13. Find the greatest number that will divide 445, 572 and 699 leaving remainders 4,5 and 6 respectively 14. Prove that every positive even integer is of the form 2q and that every positive odd integer is of the form 2q+1 15. Show that any positive odd integer is of the form 4q+1 or 4q+3 16. Show that any positive odd integer is of the form 6q+1 or 6q+3 or 6q+5 17. Use Euclid’s division lemma to show that square of any positive integer is of the form 3m or 3m+1 18. Prove that the cube of any positive integer is of the form 9m or 9m+1 or 9m+8 19. Show that the square of an odd integer is of the form 4q+1 for some integer q. 20. Show that n2 – 1 is divisible by 8, if n is an odd positive integer.
21. Show that square of any positive integer cannot be of the form 5q+2 or 5q+3 22. For any positive integer n, prove that n3 – n is divisible by 6 23. Prove that one of every three consecutive positive integers is divisible by 3 24. Show that one and only one of the numbers n, n+2 and n+4 is divisible by 3. 25. If c is a factor of q2, is c a factor of q3? Explain 26. Express 468 as product of its prime factors 27. Prime factorize 7325 28. Find the LCM and HCF of 510 and 92 using fundamental theorem of arithmetic and verify that LCMXHCF = product of the two numbers 29. Find the LCM and HCF of 144, 180 and 192 by applying the prime factorization method 30. If the prime factorizations of two numbers p and q are a3b2 and a4bc2 the find HCF(p,q) and LCM(p,q) 31. Find the HCF of 96 and 404 by prime factorization method. Hence find their LCM 32. If the HCF of 306 and 657 is 9, find their LCM 33. The LCM of two numbers is 192 and their product is 3072. Find their HCF 34. The LCM and HCF of two numbers are 180 and 6 respectively. If one of the numbers is 30 find the other. 35. 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. If 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? 36. Find the smallest number divisible by both 520 and 468 37. The length, breadth and height of a room are 8 m 25cm, 6m 75cm and 4m 50cm respectively. Determine the longest rod which can measure the three dimensions of the room exactly. 38. Two brands of chocolates are available in packs of 24 and 15 respectively. If I need to buy an equal number of chocolates of both kinds, what is the least number of boxes of each kind I would need to buy? 39. Find the greatest 6 digit number exactly divisible by 24, 15 and 36 40. What is the smallest member That, when divided by 35, 56 and 91 leaves w remainder of 7 in each case
41. A rectangular courtyard is 18m 72cm long and 13m 20cm broad. It is to be paved with square tiles of the same size. Find the least possible number of such tiles. 42. In a morning walk three persons step off together, their steps measuring 80cm, 85cm and 90cm respectively. What is the minimum distance each should walk so that it is covered in complete steps? 43. Can two numbers have 16 as their HCF and 380 as their LCM? Give reasons. 44. Can 4n end with 0 for any value of n? Give reason 45. Check whether 6n can end with the digit 0 for any natural number n 46. Show that 12n cannot end with digit o or 5 for any natural number n 47. How many zeros will be there at the end of a number if its prime factorization is 23 x 34 x 52 48. Explain why 7×11×13 +13 is a composite number 49. Is 3×5×7+7 a prime number or a composite number? Explain 50. Prove that √2 is irrational 51. Prove that √3 is irrational 52. Prove that √5 is irrational 53. Prove that √7 is irrational 54. Prove that √p is irrational where p is a prime number. 55. prove that 3+2√5 is irrational 56. Prove that 1/√2 is irrational 57. Prove that 6 + √2 is irrational 58. Show that 7√5 is irrational 59. Prove that √2+√5 is irrational 60. Show that 𝑛 − 1 + 𝑛 + 1 is an irrational number. 61. Write the following decimals as rational numbers. (a) 0.38 (b) 0.0013 (c) 1.23 (d) 0.134 62. Without performing long division, state whether the following rational numbers have a terminating decimal expansion on a non-terminating decimal expansion. Also write the decimal expansions of the ones which have a terminating decimal expansion a)
13 3125
b)
17 8
c)
64 455
d)
15 1600
e)
29 343
f)
23 200
g)
6 15
63. After how many digits will the decimal representation of their decimal representations.
35
,
453
50 1250
𝑎𝑛𝑑
47 625
terminate? Write