Real Numbers

SHORT ANSWER TYPE QUESTIONS [2 Marks]

**1.**Show that every positive even integer is of the from 2

*m*, and that every positive odd integer is of the form 2

*m*+ 1, where

*m*is some integer.

**2.**Show that any positive odd integer is of form 4

*m*+ 1 or 4

*m*+ 3, where

*m*is some integer.

**3.**Show that any positive odd integer is of the form 6

*m*+ 1, or 6

*m*+ 3, or 6

*m*+ 5, where

*m*is some integer.

**4.**Find the HCF of 1656 and 4025 by Euclid’s method.

**5.**Factorise 34650 using factor tree.

**6.**Find the HCF of 255 and 867 by prime factorisation.

**7.**Find the largest number which can divide 3528 and 2835.

**8.**Find the LCM of 2520 and 2268 by prime factorisation.

**9.**Find the smallest number which is divisible by 85 and 119.

**10.**Show that 5 - Ö3 is irrational.

**11.**Show that 3Ö2 is irrational.

**12.**Show that 1 / Ö2 is irrational.

**13.**Write the denominator of the rational number 257 / 5000 in the form 2

*× 5*

^{m}*, where*

^{n}*m, n*are non-negative integers. Hence, write its decimal expansion, without actual division.

**14.**The values of the remainder

*r,*when a positive integer

*a*is divided by 3, are 0 and 1 only. Is it true? Justify your answer.

**15.**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.

**16.**Show that the sum and product of two irrational numbers (5 + Ö2) and (5 – Ö2) are rational numbers.

**17.**Without actually performing the long division, find if 987 / 10500 will have terminating or non-terminating repeating decimal expansion. Give reason for your answer.

**18.**Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.

**19.**Show that any positive integer is of the form 3

*q*or 3

*q +*1 or 3

*q +*2 for some integer

*q*.

**20**Can the number 6

*n*,

*n*being a natural number, end with the digit 5? Give reasons.

**21.**Use Euclid’s division lemma to show that square of any positive integer is either of form 3

*m*or 31

*m*+ for some integer

*m.*

**22.**Find the L.C.M. of 120 and 70 by fundamental theorem of Arithmetic.

**23.**Write 60 in form of factor tree.

**24.**Without actually performing the long division, state whether the following number has a terminating decimal expansion or non terminating recurring decimal expansion 543 / 225.

**25.**Use Euclid’s division algorithm to find HCF of 870 and 225.

**26.**Check whether 6

*can end with the digit 0, for any natural number*

^{n}*n.*

**27.**Explain why 11 × 13 × 15 × 17 + 17 is a composite number.

**28.**Show that every positive even integer is of the form 2

*q*and that every positive odd integer is of the form 2

*q +*1, where

*q*is some integer.

**29.**Check whether 15

*n*can end with digit zero for any natural number

*n.*

**30.**Find the LCM of 336 and 54 by prime factorisation method.

**31.**Find the LCM and HCF of 120 and 144 by fundamental theorem of arithmetic.

**32.**Use Euclid’s Lemma to show that square of any positive integer is of form 4

*m*or 41

*m*+ for some integer

*m*.

**33.**Using fundamental theorem of arithmetic, find the HCF of 26, 51 and 91

**34**. Find the LCM and HCF of 15, 18, 45 by the prime factorisation method..

**35.**Find the HCF and LCM of 306 and 54. Verify that HCF × LCM = Product of the two numbers

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