Number Theory Test 10

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Number Theory Test 10

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Weekly Quiz Competition

Question 1
1 / -0

The smallest 3 digit prime number is:

A
$$101$$

B
$$103$$

C
$$109$$

D
$$113$$

Solution

The smallest 3-digit number is $$100$$, which is divisible by $$2$$.
$$\therefore$$ $$100$$ is not a prime number.
$$\sqrt{100}< 11$$ and $$101$$ is not divisible by any of the prime numbers $$2,3,5,7,11$$.
$$\therefore$$ $$101$$ is a prime number.
Hence $$101$$ is the smallest 3-digit prime number.
Question 2
1 / -0

What is the value of $$x$$ for which $$x, x + 1, x + 3$$ are all prime numbers?

A
$$0$$

B
$$1$$

C
$$2$$

D
$$101$$

Solution

Option $$A :$$ Substitute $$x=0,$$
$$x,x+1,x+3=0,1,3$$ are not prime numbers.

Option $$B :$$ Substitute $$x=1,$$
$$x,x+1,x+3=1,2,4$$ are not prime numbers.

Option $$C :$$ Substitute $$x=2,$$
$$x, x + 1, x + 3=2,3,5$$ which are all prime numbers.

Hence, option $$C$$ is correct.
Question 3
1 / -0

Which of the following number is a prime?

A
$$667$$

B
$$861$$

C
$$481$$

D
$$331$$

Solution

$$667$$ is divisible by $$23$$
$$861$$ is divisible by $$3$$
$$481$$ is divisible by $$13$$
$$331$$ is divisible only by $$1$$ and $$331$$.
$$\therefore 331$$ is a prime number.
Question 4
1 / -0

What is the modulus of $$\cfrac { \sqrt { 2 } +i }{ \sqrt { 2 } -i } $$ where $$i=\sqrt { -1 } $$

A
$$3$$

B
$$\dfrac{1}{2}$$

C
$$1$$

D
None of the above

Solution

$$\left| \dfrac { \sqrt { 2 } +i }{ \sqrt { 2 } -i } \right| =\dfrac { \sqrt { ({ \sqrt { 2 } })^{2}+{ 1 }^{ 2 } } }{ \sqrt { (\sqrt { 2 })^{ 2 }+(-1)^{ 2 } } } =1$$
Hence, option C is correct.
Question 5
1 / -0

Consider the following numbers.
1. $$247$$
2. $$203$$
Which of the above number is/are prime?

A
$$1$$ only

B
$$2$$ only

C
Both $$1$$ and $$2$$

D
Neither $$1$$ nor $$2$$

Solution

A number which is divisible by itself and $$1$$ are prime numbers.
1. $$247 = 13\times 19$$
2. $$203 = 7\times 79$$
Here both the numbers have some divisors.
So, none of the two are prime.
Question 6
1 / -0

Which one of the following is a prime number?

A
$$184$$

B
$$171$$

C
$$173$$

D
$$221$$

Solution

$$184 = 23\times 8$$
Therefore, cannot be prime
$$171 = 19\times 9$$
Therefore,cannot be prime
$$221 = 17\times 13$$
Therefore, cannot be prime
Hence, by elimination we have $$173$$ as a prime number.
Question 7
1 / -0

If $$\cos { \left( \log { { i }^{ 4i } } \right) } =a+ib$$, then

A
$$a=1,b=-1$$

B
$$a=-1,b=1$$

C
$$a=1,b=0$$

D
$$a=1,b=2$$

Solution

Given, $$a+ib=\cos { \left( \log { { i }^{ 4i } } \right) } $$
$$=\cos { \left[ 4i\log { i } \right] } =\cos { \left[ 4i\log { \left( { e }^{ i\cfrac { \pi }{ 2 } } \right) } \right] } =\cos { \left[ \left( 4i \right) \left( i\cfrac { \pi }{ 2 } \right) \right] } =\cos { \left( -2\pi \right) } =1$$
$$\therefore a=1;b=0$$
Question 8
1 / -0

Which of the following numbers is prime?

A
$$119$$

B
$$187$$

C
$$247$$

D
$$179$$

Solution

$$119$$ is divisible by $$7$$
$$187$$ is divissible by $$11$$
$$247$$ is divisible by $$13$$
Bt $$179$$ is not divisible by any number.
Hence $$179$$ is a prime number.
Question 9
1 / -0

$$( 7 \times 11 \times 13 + 13 )$$ is a/an

A
Composite number

B
Prime number

C
Irrational number

D
Imaginary number.

Solution

$$(7\times 11\times 13+13)=13\{(7\times11)+1\}=13\times 78=1014$$
Since, $$1014$$ is divided by $$13,78$$ which is other than $$1$$ and the number itself, so it is a composite number.
Question 10
1 / -0

The reciprocal of the smallest prime number is _______.

A
$$0$$

B
$$\dfrac {1}{2}$$

C
$$1$$

D
$$2$$

Solution

Prime no. are those which has only two factors.
One and the number itself.
$$Note:$$ $$1$$ is neither prime nor composite.
So, the smallest prime number is $$2$$.
Hence, reciprocal of $$2$$ $$= \dfrac{1}{2}$$

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Number Theory Test 10

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Number Theory Test 10

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SHARING IS CARING

Question 1

$$101$$

$$103$$

$$109$$

$$113$$

Solution

Question 2

$$0$$

$$1$$

$$2$$

$$101$$

Solution

Question 3

$$667$$

$$861$$

$$481$$

$$331$$

Solution

Question 4

$$3$$

$$\dfrac{1}{2}$$

$$1$$

None of the above

Solution

Question 5

$$1$$ only

$$2$$ only

Both $$1$$ and $$2$$

Neither $$1$$ nor $$2$$

Solution

Question 6

$$184$$

$$171$$

$$173$$

$$221$$

Solution

Question 7

$$a=1,b=-1$$

$$a=-1,b=1$$

$$a=1,b=0$$

$$a=1,b=2$$

Solution

Question 8

$$119$$

$$187$$

$$247$$

$$179$$

Solution

Question 9

Composite number

Prime number

Irrational number

Imaginary number.

Solution

Question 10

$$0$$

$$\dfrac {1}{2}$$

$$1$$

$$2$$

Solution

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