Real Numbers

what is real no ?

 Prove that root 5 is irrational.

Show that one and only one out of n,n+2,n+4 is divisible by 3, where n is any positive integer.

Smallest odd composite number and largest two digit prime number

Any positive odd integer is of the form 4q+1 or 4q+3

The sets of mathematics, physics and physical education books have to be stacked
in such a way that all the books are stored topic wise. The number of mathematics
, physics and physical education books are 14 , 18 and 22. Determine the number
of stacks of each books provided books are of the same thickness.

In a morning walk Nirmaljeet, Puneet, Rajiv step off together, their steps
measuring 240 cm , 90 cm, 120cm respectively. What is the minimum distance
each should walk so that one can cover the distance in complete steps?

Find the largest positive integer that will divide 122,150 and 115 leaving
remainder 5, 7 and 11 respectively.

Use Euclid‟s division algorithm to find the largest number which divides 957 and
1280 leaving remainder 5 in each case.

Find the HCF of 616 and 32 using Euclid’s division algorithm. 

Explain why ( 7 x 11 x 13) + 13 is a composite number. 

Show that 3√2 is irrational.

Find the LCM and HCF of 510 and 92 and verify the thatLCMxHCF = Product of
two no.

Show that the square of any positive integer is of the form 3m or 3m+1,for some
integer m.

Find the LCM and HCF of 12 , 15 and 21 by prime factorization method.

The HCF of two number is 4 and their LCM is 9696,if one number is 96,find the
other number.

Find the LCM and HCF of 91 and 26, also verify that HCF×LCM= product of
two

Explain why 7×6×5×4×3×2×1+5 is a composite number.

 Find HCF of 128 and 216. 

The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form  , what can you say about the prime factor of q?

(i) 43.123456789 (ii) 0.120120012000120000… (iii) 

Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.

Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

Prove that the following are irrationals:

Prove that  is irrational.

Prove that  is irrational.

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. 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?

Check whether 6n can end with the digit 0 for any natural number n.

Given that HCF (306, 657) = 9, find LCM (306, 657).

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

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.

Express each number as product of its prime factors:

Use Euclid's division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m +8.

Use Euclid's division lemma to show that the square of any positive integer is either of form 3m or 3m + 1 for some integer m.

[Hint: Let x be any positive integer then it is of the form 3q, 3q + 1 or 3q + 2. Now square each of these and show that they can be rewritten in the form 3m or 3m + 1.]

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 the same number of columns. What is the maximum number of columns in which they can march?

Show that any positive odd integer is of the form , or , or , where q is some integer.

Use Euclid's division algorithm to find the HCF of: