Real Numbers

Q.

find the smallest Number which leaves remainder 8 and 12 when divided by 28 and 32 respectively

Q.

Determine the greatest 3-digit number exactly divisible by 8, 10 and 12.

Q. Find five rational number between 4 and 5

Q. Prove that the square of any positive integer is of the form 3m or, 3m + 1 but not of the form 3m + 2.

Q.

Show that the square of any positive integer is either of the form 4q or 4q+1, for some integer q.

Q. Find the three rational numbers between 1/4 and 1/2

Q.

Prove that $n^2-n$ is divisible by 2 for any positive integer n

Q.

show that square of an odd positive integer is of the form 8m+1, for some whole number m

Q. Of the given pairs of numbers, the pair(s) of co-prime numbers is/are:
(i) 231, 396
(ii)847, 2160

1. Only (i)
2. Only (ii)
3. Both (i) and (ii)
4. Neither (i) nor (ii)

Q.

Show that the square of an odd positive integer can be of the form $6q+1$ or $6q+3$ for some integer $q$.

Q. Find 2 Rational Numbers between 1/3 and 2/5

Q.

Show that square of an odd integer is of the form 4q+1 for some integer q

Q. Show that any positive odd integers is of the form 4q+1 or 4q+3 where q is a positive integer.

Q. Question 4
Use Euclid's division lemma to show that the square of any positive integer is either of the 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.]

Q. Find the HCF , by Euclid's division algorithm of the numbers 92690, 7378 and 7161

Q. Without actually performing the long division, state whether $\frac{6}{15}$ will have a terminating decimal expansion or a non-terminating repeating decimal expansion.

Q.

Show that any positive even integer can be written in the form 6q, 6q+2 and 6q+4 where q is an integer

Q. Show that the square of an odd positive integer is of the form 8q + 1, for some integer q.

Q. Find four rational number between 4 and 5. How many rational number lie between these two numbers.

Q. The difference between any two rational numbers is always a .

1. natural number
2. rational number
3. whole number

Q.

show that the square of any positive integer cannot be of the form 5m+2 or 5m+3 for any integer m

Q. Euclid's division lemma states that for two positive integers a and b , there exist unique integers q and r such that a = bq + r , where r must satisfy
(a) 1 < r < b (b) 0 < r $\le$ b (c) 0 $\le$ r < b (d) 0 < r < b r

Q.

Use Euclid Division algorithm to find the HCF of 1620, 225, 1725.

Q.

Show that square of any positive integer cannot be of the form 6m+2 or 6m+5 for any integer m

Q.

Is root3/2 a rational or irrational number.

Q. The sum of two irrational number is an irrational number (True/False).

Q. For any integers a and 3, there exists unique integers q and r such that a=3q+r. Find the possible value of r.

Q.

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

Q. Find the LCM and HFC of the following pairs of integers using fundamental theorem of arithmetic and verify that LCM×HFC= product of the 2 numbers i) 78 and 117

Q. Find the HCF of the positive integers 180, 252 and 324 with the help of division algorithm.