Number Theory Test 4

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Number Theory Test 4

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Weekly Quiz Competition

Question 1
1 / -0

Which of the following is a prime number?

A
$$889$$

B
$$997$$

C
$$899$$

D
$$1147$$

Solution

A prime can be written as $$6k+1$$ or $$6k-1$$ provided the number is greater than $$6$$.
$$997=166*6+1$$
Therefore, $$997$$ is a prime number.
Question 2
1 / -0

Write all prime numbers between $$20$$ and $$50.$$

A
$$21, 29, 33, 37, 41, 43$$ and $$47$$

B
$$29, 33, 37, 39, 41, 43$$ and $$47$$

C
$$23, 29, 31, 37, 41, 43$$ and $$47$$

D
None of these

Solution

All prime numbers between $$ 20 $$ and $$ 50 $$ are $$

23, 29, 31, 37, 41, 43 $$ and $$ 47. $$
Question 3
1 / -0

Solve $$\displaystyle \left ( 1-i \right )x+\left ( 1+i \right )y= 1-3i,$$

A
$$\displaystyle x= -1, y= 2.$$

B
$$\displaystyle x= 2, y= -1.$$

C
$$\displaystyle x= 2, y= 1.$$

D
$$\displaystyle x= 1, y= 2.$$

Solution

$$\displaystyle \left ( 1-i \right )x+\left ( 1+i \right )y= 1-3i.$$
Equating real and imaginary parts, we get
$$\displaystyle x+y= 1$$ and $$\displaystyle -x+y= -3.$$
Adding both equations we get $$y=-1$$
Substituting this value of $$y$$ in any of the 2 equations, we get $$x=2$$
$$\therefore \displaystyle x= 2, y= -1.$$

Ans: B
Question 4
1 / -0

Evaluate :
$$\sqrt{-25} + 3 \sqrt{-4} +2 \sqrt{-9}$$

A
$$-17i$$

B
$$5i$$

C
$$17i$$

D
$$6i$$

Solution

$$\sqrt{-25} + 3 \sqrt{-4} +2 \sqrt{-9} $$

$$=5\sqrt{-1}+6\sqrt{-1}+6\sqrt{-1}$$ we know, $$\sqrt{-1}=i$$

$$= 5i + 6i + 6i = 17i$$
Question 5
1 / -0

How many prime numbers are there between $$0$$ and $$30$$?

A
$$9$$

B
$$10$$

C
$$8$$

D
$$11$$

Solution

Prime numbers:- The numbers which are divisible by $$1$$ and themselves, or the number which has only two factors, $$1 $$ and the number itself
e.g. $$7$$
Note:- $$ 1$$ is not a prime number, the smallest prime number is $$2$$
Prime numbers between $$1$$ and $$30$$ are $$2,~3,~5,~7,~11,~13,~17,~19,~23,~29.$$
Total ten numbers are there.
Question 6
1 / -0

An example for twin primes is

A
$$5, 11$$

B
$$3, 5$$

C
$$11, 17$$

D
$$3, 7$$

Solution

Twin primes are those prime which has only one even number between them.
Out of given options clearly $$3$$ and $$5$$ are twin primes as only even number $$4$$ lie between them.
Option $$B$$ is correct.
Question 7
1 / -0

The number which is neither prime nor composite is

A
$$3$$

B
$$1$$

C
$$2$$

D
$$5$$

Solution

A prime number is a number which has only two factors, $$1$$ and the number itself. So, a prime number has 2 factors.
A composite number is a number that has factors other than $$1$$ and the number itself. A composite number has more than 2 factors.
The number which is neither prime nor composite is $$ 1 $$ as it is divisible only by $$1$$. It has only one factor.
Hence, Option B is correct answer.
Question 8
1 / -0

The two consecutive prime numbers with difference $$2$$ are called

A
co-primes

B
twin primes

C
composite

D
none of these

Solution

Twin prime numbers are two prime numbers whose difference is $$2.$$
Example : $$3$$ and $$5,$$ $$17$$ and $$19 ,41$$ and $$43$$ etc.
Question 9
1 / -0

Number of prime numbers from 1 to 50 are

A
$$18$$

B
$$12$$

C
$$15$$

D
$$20$$

Solution

There are 15 prime numbers from 1 to 50. They are $$2, 3, 5, 7, 11, 13, 17,19, 23, 29, 31,37,41,43,47.$$
Question 10
1 / -0

The numbers which have more than two factors are called

A
even

B
prime

C
composite

D
none of these

Solution

The numbers which have more than two factors are called composite numbers.

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Number Theory Test 4

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Number Theory Test 4

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Question 1

$$889$$

$$997$$

$$899$$

$$1147$$

Solution

Question 2

$$21, 29, 33, 37, 41, 43$$ and $$47$$

$$29, 33, 37, 39, 41, 43$$ and $$47$$

$$23, 29, 31, 37, 41, 43$$ and $$47$$

None of these

Solution

Question 3

$$\displaystyle x= -1, y= 2.$$

$$\displaystyle x= 2, y= -1.$$

$$\displaystyle x= 2, y= 1.$$

$$\displaystyle x= 1, y= 2.$$

Solution

Question 4

$$-17i$$

$$5i$$

$$17i$$

$$6i$$

Solution

Question 5

$$9$$

$$10$$

$$8$$

$$11$$

Solution

Question 6

$$5, 11$$

$$3, 5$$

$$11, 17$$

$$3, 7$$

Solution

Question 7

$$3$$

$$1$$

$$2$$

$$5$$

Solution

Question 8

co-primes

twin primes

composite

none of these

Solution

Question 9

$$18$$

$$12$$

$$15$$

$$20$$

Solution

Question 10

even

prime

composite

none of these

Solution

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