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Real Numbers

Short Answer Type

101. Find the HCF and LCM of 144, 180 and 192 by prime factorisation method.

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102. Find the largest positive integer that will divide 398, 436 and 542 having remainders 7. 11 and 15 respectively.

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103. In a seminar, the number of participants in physics, chemistry 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.

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104.

A rectangular garden is 18 m 72 cm long and 13 m 20 cm broad. It is to be paved with square tiles of the same size. Find the least possible number of such Files.

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105.

Find the LCM and HCF of the following integers by applying the prime factorisation method :

(i) 40. 36 and 126 (ii) 24, 15 and 36

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106. There are 68 boys and 76 girls in class X of a school. The principal of school decided to have the maximum number of mixed sections in which the number of boys and number of girls should remain the same in all sections. It is not necessary that there is equal number of boys and girls in a section. What is the maximum number of sections in the school.

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107. A circular field has a circumference of 360 km. Three cyclists start together and can cycle 48. 60 and 72 km a day, round the Field. When will they meet again.

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108. Express 0.2317 in the form

straight p over straight q

where p and q are co-prime and also express q in the form 2² × 5^m.

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109. Express 0.737373.......in the form

straight P over straight q

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110.
Prove that $open parentheses 5 minus 2 square root of 3 close parentheses$ is an irrational number.

Let us assume, to the contrary, that

open parentheses 5 minus 2 square root of 3 close parentheses

is rational i.e. we can find coprimes a and b (b ≠ 0) such that

5 minus 2 square root of 3 equals straight a over straight b rightwards double arrow 2 square root of 3 equals 5 minus straight a over straight b equals fraction numerator 5 straight b minus straight a over denominator straight b end fraction rightwards double arrow space space square root of 3 equals fraction numerator 5 straight b minus straight a over denominator 2 straight b end fraction

Since a and b are integers, we get

fraction numerator 5 straight b minus straight a over denominator 2 straight b end fraction

is rational, and so

space space square root of 3 space

is rational
BUt thisc contradicts the fact that

square root of 3

is irrational.

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