Number Theory Test 3

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Number Theory Test 3

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Weekly Quiz Competition

Question 1
1 / -0

Which of the following is not a composite number?

A
$$352+6$$

B
$$357+7$$

C
$$352+7$$

D
$$353+1$$

Solution

$$A:$$ $$352+6 = 358$$ which is even, and have factors other than $$1$$ and itself, So it is a Composite number.
$$B:$$ $$357+7 = 364$$ which is even, and have factors other than $$1$$ and itself, So it is a Composite number.
$$C:$$ $$352+7 = 359$$ which is odd, and does not have factors other than $$1$$ and itself, So it is a Prime number.
$$D:$$ $$353+1 = 354$$ which is even, and have factors other than $$1$$ and itself, So it is a Composite number.

Hence, option $$C.$$
Question 2
1 / -0

The total number of prime numbers between $$120$$ and $$140$$ is

A
$$7$$

B
$$6$$

C
$$5$$

D
$$4$$

Solution

The prime numbers must be from the odd numbers between $$120$$
and $$140$$.
The prime numbers are $$127, 131, 137, 139$$.
Therefore, there are 4 prime numbers between $$120$$ and $$140$$.
Question 3
1 / -0

If the square of $$(a + ib)$$ is real, then $$ ab=$$

A
$$0$$

B
$$1$$

C
$$-1$$

D
$$2$$

Solution

$$(a+ib)^{2}=a^{2}-b^{2}+2iab$$ is given to be real
$$\Rightarrow ab=0$$
Question 4
1 / -0

Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique apart from the order in which the prime factors occur.

A
True

B
False

C
Neither

D
Either

Solution

This is true as any composite number can be expressed in terms of its prime factors. The prime factors for 2 numbers can be same but their combination is unique.
Example,
$$12=2^2*3$$ and $$18=2*3^2$$
Both have prime factors as 2 and 3 but the combination that 12 has is different from that of 18.
Question 5
1 / -0

Which of the following is not a prime number ?

A
$$2$$

B
$$3$$

C
$$4$$

D
$$5$$

Solution

The factors of $$4$$ are $$1$$, $$2$$ and $$4$$.
As it has a factor other than itself and $$1$$, it is not a prime number.
Question 6
1 / -0

The number of composite numbers between $$101$$ and $$120$$
(excluding both) are

A
$$11$$

B
$$12$$

C
$$13$$

D
$$14$$

Solution

There are $$9$$ even numbers which are all composite numbers.
Amongst the odd numbers $$105, 111, 115, 117, 119$$ are composite numbers.
So, there are $$14$$ composite numbers in the given range.
Question 7
1 / -0

What is the remainder obtained when a prime number greater than $$6$$ is divided by $$6$$?

A
$$1 \ or\ 3$$

B
$$1 \ or\ 5$$

C
$$3\ or\ 5$$

D
$$4 \ or\ 5$$

Solution

We can take an example. Prime numbers greater than $$6$$ are $$7, 11, 13,$$ and so on. When $$7$$ is divided by $$6,$$ remainder is $$1$$. When $$11$$ is divided by $$6$$, remainder is $$5$$. Similarly, when $$13$$ is divided by $$6$$, remainder is $$1$$. The remainder obtained is either $$1$$ or $$5$$.
So the correct answer will be option $$B$$
Question 8
1 / -0

A number other than one which is either divisible by $$1$$ or itself is called a

A
composite number

B
prime number

C
comprime number

D
none of these

Solution

Given that a number is divisible by itself and $$1$$.
$$\therefore$$ If x is the number then its factors are x and $$1$$. So this fits in the condition for a number to be prime.
$$\therefore$$ It is a prime number.
Question 9
1 / -0

Write two pairs of twin primes between $$20$$ and $$50.$$

A
$$16$$ and $$26, 19$$ and $$29$$

B
$$29$$ and $$31, 41$$ and $$43$$

C
$$11$$ and $$21, 61$$ and $$71$$

D
$$15$$ and $$17, 37$$ and $$39$$

Solution

A twin prime is a set of prime numbers that must be either $$2$$ less or $$2$$ more than another prime number. Example $$5$$ and $$7, 11$$ and $$13.$$

$$29$$ and $$31, 41$$ and $$43$$ are two pairs of twin primes between $$20$$ and $$50.$$
Question 10
1 / -0

Find the value of $$x$$ of the equation $${ \left( 1-i \right) }^{ x }={ 2 }^{ x }$$

A
$$1$$

B
$$2$$

C
$$0$$

D
none of these

Solution

Then $${ \left( 1-i \right) }^{ x }={ 2 }^{ x }$$

$$\Rightarrow |{(1-i)}|^{x} =|2|^{x}$$

$$\Rightarrow { \left( \sqrt {1+1}\right)}^{x}={2}^{x}$$

$$\Rightarrow { \left( \sqrt { 2 } \right) }^{ x }={ 2 }^{ x }$$

$$\Rightarrow \dfrac{x}{2}=x$$

$$\Rightarrow 2x=x\Rightarrow x=0$$
Hence, option C is correct.

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Number Theory Test 3

Result

Number Theory Test 3

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SHARING IS CARING

Question 1

$$352+6$$

$$357+7$$

$$352+7$$

$$353+1$$

Solution

Question 2

$$7$$

$$6$$

$$5$$

$$4$$

Solution

Question 3

$$0$$

$$1$$

$$-1$$

$$2$$

Solution

Question 4

True

False

Neither

Either

Solution

Question 5

$$2$$

$$3$$

$$4$$

$$5$$

Solution

Question 6

$$11$$

$$12$$

$$13$$

$$14$$

Solution

Question 7

$$1 \ or\ 3$$

$$1 \ or\ 5$$

$$3\ or\ 5$$

$$4 \ or\ 5$$

Solution

Question 8

composite number

prime number

comprime number

none of these

Solution

Question 9

$$16$$ and $$26, 19$$ and $$29$$

$$29$$ and $$31, 41$$ and $$43$$

$$11$$ and $$21, 61$$ and $$71$$

$$15$$ and $$17, 37$$ and $$39$$

Solution

Question 10

$$1$$

$$2$$

$$0$$

none of these

Solution

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