Number Theory: Interesting Results

Q.

Question 14
After rationalizing the denominator of $\frac{7}{3\sqrt{3}-2\sqrt{2}}$, we get the denominator as

A) 13
B) 19
C) 5
D) 35

Q. Question 2
Simplify:
$\frac{7\sqrt{3}}{\sqrt{10}+\sqrt{3}}-\frac{2\sqrt{5}}{\sqrt{6}+\sqrt{5}}-\frac{3\sqrt{2}}{\sqrt{15}+3\sqrt{2}}$

Q.

Find which of the variables $x$, $y$, $z$and $u$ represent rational numbers and which irrational numbers:

$x^2=5$

Q.

The square root of a negative number is not a real number. Why?

Q. Prove the following number is irrational.
$\frac{1}{\sqrt{2}}$

Q. Question 18
The product $\sqrt[3]{32}.\sqrt[4]{42}.\sqrt[12]{1232}$ equals to
Thinking process
Take the LCM of indices of three irrational numbers. Then, convert all individually in the from whose index will be equal to LCM.

A) $\sqrt{2}$
B) $2$
C) $\sqrt[12]{122}$
D) $\sqrt[12]{1232}$

Q.

If x and y are the two digits of the number 653xy such that this number is divisible by 80, then x + y = ?

1. 8

2. 4 or 8

3. 6

4. 4

Q.

If $\sqrt{10}=3.162$ then find the value of $\frac{1}{\sqrt{10}}$.

Q.

An n-digit number is a positive number with exactly n digits. Atleast nine hundred distinct n-digit numbers are to be formed using only the three digits 2, 5 and 7. The smallest value of n for which this is possible is

1. 7

2. 8

3. 6

4. 9

Q.

Is Pi $\left(\pi \right)$ a Real Number?

Q. Given that $\sqrt{3}$ is an irrational number, prove that $(2+\sqrt{3})$ is an irrational number.

Q. Prove that $\sqrt{2}$ is irrational and hence prove that $\frac{5-3\sqrt{2}}{7}$ is irrational.

Q. Prove that $\sqrt{6}$ is an irrational number

Q.

Every natural number is a

1. None

2. Integer

3. Rational number

4. Fraction

Q.

The square of every natural number is always greater than the number itself. State whether the statement is true (T) or false (F).

1. True
2. False

Q. If $x=2+\sqrt{3},xy=1,$ then find $\frac{x}{2-x}+\frac{y}{2-y}$.

1. $2\sqrt{3}$
2. $\frac{4}{\sqrt{3}}$
3. $-3$
4. $-2$

Q.

For an integer n, a student states the following:
I. If $n$ is odd, $(n+1{)}^2$ is even
II. If $n$ is even, $(n-1{)}^2$ is odd
III. If $n$ is even, $\sqrt{n-1}$ is irrational.
Which of the above statements would be true?

1. I and III

2. I and II

3. I, II and III

4. II and III

Q. Prove that $\sqrt{2}$ is on irrational number and also prove that $3+5\sqrt{2}$ is irrational number.

Q. Prove that $5+\surd 2$ is an irrational number,

Q.

Prove that $\sqrt{7}$ is irrational

Q. Prove that $\sqrt{3}$ is an irrational number. Hence, show that $7+2\sqrt{3}$ is also an irrational number.

Q. If $k_1,k_2,k_3,...,k_n$ are odd natural numbers, then the remainder when $k_1^2+k_2^2+k_3^2+...+k_n^2$ is divided by 4, is always equal to

Q.

Find the square root of $90$ in radical form.

Q. Given that $\sqrt{2}$ is irrational, prove that $(5+3\sqrt{2})$ is an irrational number.

Q. The product of number of odd and even factors of 36000 will be

1. 100
2. 864
3. 720
4. 72

Q. For an integer m, the square of any positive integer cannot be of the form

1. 6m + 4
2. 5m + 4
3. 4m + 1
4. 6m + 5

Q. Prove that $\sqrt{5}$ is an irrational number.

Q. Which of the following numbers are irrational numbers?

1. $2-\sqrt{5}$
2. $\left(3+\sqrt{23}\right)-\left(\sqrt{23}\right)$
3. $\frac{1}{\sqrt{2}}$
4. $2\pi$

Q. Show that there is no positive integer n for which $\sqrt{n-1}+\sqrt{n+1}$ is rational.

Q.

Rationalise and simplify the expression
$\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{2}-\sqrt{5}}+\frac{2}{\sqrt{5}-\sqrt{3}}$

1. $1+\sqrt{5}$

2. $\sqrt{2}+\sqrt{3}+\sqrt{5}$

3. $1+\sqrt{3}$

4. 0