1. The document discusses properties of real numbers including: Euclid's division lemma, the fundamental theorem of arithmetic, unique prime factorizations of composite numbers, properties of prime numbers, laws of logarithms, and multiple choice questions testing knowledge of real number concepts. 2. Key real number concepts covered include: every positive integer can be uniquely expressed as a product of prime factors; there are infinitely many prime numbers; rational numbers may have terminating or repeating decimal expansions based on their prime factorizations; laws of logarithms include log(ab) = loga + logb. 3. The multiple choice questions test understanding of concepts like: determining if a number is rational or irrational; finding highest common factors and

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1.Real Numbers

2. 1.Real Numbers
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Real Numbers
1) Euclid’s Division Lemma:
Given Positive integers and there exists whole numbers and satisfying
2) The Fundamental theorem of arithmetic :
Every composite number can be expressed as a product of primes, and this
factorization is unique except for the order in which the prime factors occur.
3) Every composite number can be uniquely expressed as the product of powers of
primes in ascending or descending order.
4) Let be a positive integer and be a prime number such that divides then
divides .
5) There are infinitely many positive prime numbers,
6) Every positive integer different from 1 can be expressed as product of non
negative powers of 2 and an odd number
7) A positive integer “n” is prime if it is not divisible by any prime less than or equal
to .
8) If is a positive prime, then is an irrational number. For eg. ,… etc.
are irrational numbers.
9) Let x be a rational number whose decimal expansion terminates. Then x can be
expressed in the form , where p and q are co prime, and the prime factorization
of q is of the form . When m and n are non negative integers.
10) Let be a rational number, such that the prime factorization of q is of the
form . Where m and n are non negative integers. Then has a
terminating decimal expansion which terminates after k places of decimals,
where k is larger than and .
11) Let be a rational number, such that the prime factorization of q is not of
the form where m and n are non negative integers. Then x has non
terminating repeating decimal expansion.
12) We define if , where a and x are positive numbers and

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13) Laws of logarithms
i.
ii.
iii.
iv.
v.

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Multiple Choice Questions
1. If p be a prime number. If p divides then
(a) divides (b) p divides a (c) p divides (d)None
2. The number is
(a) rational (b) irrational (c) integer (d)None
3. If p and q are prime numbers then is
(a) Irrational (b) Prime (c) rational (d)None
4. =
(a) log7-log5 (b) 2(log7-log5) (c) 3(log7-log5) (d)1
5. Number of digits in if .
(a) 1210 (b)1212 (c)1211 (d)0
6. =
(a) 11 (b)10 (c)12 (d)13
7.
(a) 0 (b)7 (c)1 (d)10
8. Last digit in
(a) 0 (b)4 (c)6 (d)None
9.
(a) 1001 (b)101 (c)11 (d)111
10. H.C.F of 306 and 657
(a) 16 (b)36 (c)9 (d)7
11. The decimal expansion of the rational number will terminate after
(a) one decimal place (b)two decimal places
(c) Three decimal places (d)four decimal places
12. The exponent of 2 in the prime factorization of 144, is
(a) 4 (b)5 (c)6 (d)3
13. If the L.C.M of and 18 is 36 and H.C.F of and 18 is 2, then
(a) 2 (b)3 (c)4 (d)1
14. . is
(a) an integer (b) rational
(c)natural number (d)irrational

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15. The remainder when the square of any prime number greater than 3 is divided
by 6, is
(a) 1 (b)3 (c)2 (d)4
16. The L.C.M and H.C.F of two rational numbers are equal then the numbers must be
(a) prime (b)Co prime (c)composite (d)equal
17. If n is any natural number, then always ends with
(a) 1 (b)3 (c)5 (d)7
18. For every positive integer “ is divisible by
(a) 3 (b)4 (c)2 (d)1
19. For every positive integer “ is divisible by
(a) 6 (b)3 (c)9 (d)None
20. If and then
(a) (b)3 (c) 2 (d) 2
21. A merchant has 120 liters of oil of one kind, 180 liters of oil of another kind and
240 liters of third kind. He wants to sell the oil by filling the three kinds of oil in
the tins of equal capacity. What should be the greatest capacity of such tin
(a) 72 liters (b)60 liters (c) 120 liters (d) None
22. If H.C.F(306,657)=9 then L.C.M(306,657)=
(a) (b)22352 (c) 22338 (d) None
23. The greatest six digit number exactly divisible by 24, 15 and 16
(a) (b)999799 (c) 999720 (d)None
24. If then
(a) 1 (b)-1 (c)0 (d)None
25. If and then L.C.M(a, b, c)= then
(a) 1 (b)2 (c)3 (d)9
26. If and being prime numbers then L.C.M(a,b)=
(a) (b) (c) (d)
27. If the sum of LCM and HCF of two numbers is 1260 and their LCM is 900 more
than their HCF, then the product of two numbers is
(a) 203400 (b)194400 (c)198400 (d)205400
28. Prime factorization of 13915 is
(a) . . (b) . . (c) . . . (d) . .

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29. The sum of the exponents of the prime factors in the prime factorization of 196,
is
(a) 1 (b)2 (c)4 (d)6
30. The HCF of 95 and 152, is
(a) 57 (b)1 (c)19 (d)38
31. If p and q are co prime numbers, then and are
(a) co prime (b)not co prime (c)even (d)odd
32. Which of the following rational number have terminating decimal
(a) (b) (c) (d)
33. The smallest rational number by which should be multiplied so that its decimal
expansion terminated after one place of decimal, is
(a) (b) (c) (d)
34. The smallest number by which should be multiplied so as to get a rational
number is
(a) (b) (c) (d)3
35. If HCF(26,169)=13, then LCM(26,169)=
(a) 26 (b)52 (c)338 (d)13
36. is
(a) Rational (b)Irrational (c)prime (d)None
37. If then
(a) 25 (b)36 (c)49 (d)None
38. If then log
(a) 2 (b)1 (c)0 (d)None
39. . =
(a) 3 (b)1 (c)-3 (d)2
40. =
(a) 1 (b)2 (c (d)0
41. H.C.F of two consecutive numbers and
(a) (b) (c) (d)

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42.
(a) (b) (c) (d)
43.
(a) (b)2 (c) (d)
44. . .
(a) (b) (c)1 (d) 3
45. then
(a)-1 (b)7 (c)2 (d) 1
46.
(a) (b) (c) (d)
47. Among the following statements which is true
(a) (b) (c) (d) None
48.
(a)-1 (b)1 (c) (d)
49. then
(a)1 (b)-1 (c) (d)
50. L.C.M of and
(a) (b) (c) (d)
Answers
1 2 3 4 5 6 7 8 9 10
b a a c b c a c a c
11 12 13 14 15 16 17 18 19 20
d a c b a d a c a d
21 22 23 24 25 26 27 28 29 30
b c c a b c b d c c
31 32 33 34 35 36 37 38 39 40
a d a c c a d d c c
41 42 43 44 45 46 47 48 49 50
d d c d c c b b b d

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