Number Theory: Interesting Results

Q.

Let p be a prime number. If p divides $a^2$ then _________, where a is a positive integer.

1. p divides a

2. a divides p

3. ap divides a

4. ap divides p

Q. Define irrational numbers with an example.

Q.

Let p be a prime number. If p divides $a^2$ then p divides a, where a is a positive integer. Which theorem is this statement previously based on?

1. Real Number theorem

2. Fundamental Theorem of Arithmetic

3. None of These

4. Euclid’s Division Lemma

Q. Rationalise the denominator of the following:

$\left(vii\right)\frac{3\sqrt{2}}{\sqrt{5}}$

Q. After rationalising the denominator of $\frac{7}{3\sqrt{3}-2\sqrt{2}}$ , we get the denominator as .

Q.

If the number $91876y2$ is completely divisible by $8$, then the smallest whole number in place of $y$ will be:

1. 1

2. 3

3. 4

4. 2

Q. The number obtained by rationalising the denominator of $\frac{1}{\sqrt{7}+2}$ is .

Q. Difference between Constructing Square root spiral and number line?

Q. Classify the following number as rational or irrational.
$5-\sqrt{3}$

Q. State whether the following statement is true or false:
All real numbers are irrational

1. True
2. False