# Prime and Composite Numbers | Algebra - Number Theory

**Prime and Composite Numbers | Algebra - Number Theory**

Product of 50 prime numbers will be

Type in your answer as

1 for even

2 for odd

3 for cannot be determined

Answer: 3

**Explanation** :

We know all prime numbers except 2 are odd.

If we multiply 50 prime numbers except 2, the product will be odd.

But if one of the prime numbers is 2, the product will be even.

Hence, 3.

Workspace:

**Prime and Composite Numbers | Algebra - Number Theory**

Sum of 51 of distinct prime number is calculated and it was found to be even. One of the prime numbers should definitely be

Answer: 2

**Explanation** :

The only even prime number is 2.

If we calculate the sum of 51 distinct prime numbers except 2, the sum will always be odd.

Sum of 51 prime numbers can be even only if one of the primes numbers is definitely 2.

Hence, 2.

Workspace:

**Prime and Composite Numbers | Algebra - Number Theory**

The sum of 75 composite numbers will be:

Type in your answer as

1 for even

2 for odd

3 for cannot be determined

Answer: 3

**Explanation** :

If all the composite numbers are even, the sum will be even.

If all the composite numbers are odd, the sum will be odd.

Hence, 3.

Workspace:

**Prime and Composite Numbers | Algebra - Number Theory**

How many prime numbers are there from 1 till 100?

Answer: 25

**Explanation** :

There are 25 prime number from 1 till 100.

[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]

Hence, 25.

Workspace:

**Prime and Composite Numbers | Algebra - Number Theory**

Which of the following is a prime number?

- (a)
391

- (b)
299

- (c)
1001

- (d)
997

Answer: Option D

**Explanation** :

Prime numbers are those numbers which have exactly 2 factors i.e., 1 and the number itself.

Option (a): 391 = 17 × 23. Factors are 1, 17, 23 and 391. Hence not a prime number.

Option (b): 299 = 13 × 23. Factors are 1, 13, 23 and 299. Hence not a prime number.

Option (c): 1001 = 7 × 11 × 13. Factors are 1, 7, 11, 13 etc.. Hence not a prime number.

Option (d): 997 is a prime number.

Hence, option (d).

Workspace:

**Prime and Composite Numbers | Algebra - Number Theory**

Which of the following are co-prime numbers?

- (a)
12 and 99

- (b)
36 and 81

- (c)
25 and 36

- (d)
None of these

Answer: Option C

**Explanation** :

Co-prime numbers are those pair of numbers which have no common factor apart from 1.

Option (a): 12 and 99 have 3 as common factor, hence they are not co-prime

Option (b): 36 and 81 have 9 as common factor, hence they are not co-prime

Option (c): 25 and 36 have no common factor apart from 1, hence they are co-prime

Hence, option (c).

Workspace:

**Prime and Composite Numbers | Algebra - Number Theory**

How many values of p are possible such p is a prime number and (p + 1) is a perfect square.

Answer: 1

**Explanation** :

Givne, p + 1 is a perfect square.

Let p + 1 = n^{2}

⇒ p = n^{2} - 1

⇒ p = (n + 1)(n - 1)

Now, if p is a prime number, the only wasy of writing p as a product of two numbers is p × 1

∴ (n + 1) = p and (n - 1) = 1

⇒ n = 2 and p = 3

∴ Only one value of p is possible.

Hence, 1.

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