Prime Numbers & Factorization MCQ Test — Class 6 CBSE

Solution:

Factors are numbers that divide a given number exactly, leaving no remainder.

Every number can be divided by 1. For example, 5 ÷ 1 = 5, 100 ÷ 1 = 100.

Therefore, 1 is a factor of every number. Options A, B, and D are incorrect based on the definitions of factors and prime numbers.

Solution:

A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself.

For 9, the factors are 1, 3, 9 (more than two factors). So, 9 is composite.

For 15, the factors are 1, 3, 5, 15 (more than two factors). So, 15 is composite.

For 21, the factors are 1, 3, 7, 21 (more than two factors). So, 21 is composite.

For 13, the factors are 1, 13 (exactly two factors). Thus, 13 is a prime number.

Solution:

A composite number is a natural number greater than 1 that has more than two factors.

Number 2 has factors 1, 2 (prime).

Number 3 has factors 1, 3 (prime).

Number 4 has factors 1, 2, 4 (more than two factors), so it is a composite number.

Number 5 has factors 1, 5 (prime).

Solution:

Prime factorization is the process of expressing a composite number as a product of its prime factors.

First, check if all factors in Ravi's expression (2, 3, 5) are prime numbers. Yes, they are.

Next, check if their product equals the original number: 2 × 3 × 5 = 6 × 5 = 30. Yes, it does.

Therefore, Ravi's prime factorization is correct. 1 is not included in prime factorization, and the order of factors does not matter.

Solution:

A prime number has exactly two distinct factors: 1 and itself. The number 1 has only one factor (1 itself).

A composite number has more than two factors. The number 1 has only one factor.

Since 1 does not fit either definition, it is classified as neither a prime nor a composite number.

Solution:

A prime number has exactly two factors: 1 and itself.

Even numbers are integers that are divisible by 2. All even numbers greater than 2 have at least three factors: 1, 2, and the number itself.

For example, 4 has factors 1, 2, 4. 6 has factors 1, 2, 3, 6.

The number 2 has factors 1 and 2. It is even and has exactly two factors, making it the only even prime number.

Solution:

To find all factors of 18, we look for pairs of numbers that multiply to 18.

1 × 18 = 18, so 1 and 18 are factors.

2 × 9 = 18, so 2 and 9 are factors.

3 × 6 = 18, so 3 and 6 are factors.

The next number to check is 4, but 18 is not divisible by 4. After 3, the next factor we found was 6, which means we have found all unique factor pairs. Listing them in order gives 1, 2, 3, 6, 9, 18.

Solution:

A pair of numbers is co-prime if their only common factor is 1.

For (4, 8): Factors of 4 are 1, 2, 4. Factors of 8 are 1, 2, 4, 8. Common factors are 1, 2, 4. Not co-prime.

For (7, 21): Factors of 7 are 1, 7. Factors of 21 are 1, 3, 7, 21. Common factors are 1, 7. Not co-prime.

For (12, 18): Factors of 12 are 1, 2, 3, 4, 6, 12. Factors of 18 are 1, 2, 3, 6, 9, 18. Common factors are 1, 2, 3, 6. Not co-prime.

For (10, 13): Factors of 10 are 1, 2, 5, 10. Factors of 13 are 1, 13. The only common factor is 1. So, (10, 13) is a co-prime pair.

Solution:

Let's analyze each statement.

A) False. Multiples are generally equal to or larger than the number (e.g., multiples of 3 are 3, 6, 9...). The smallest multiple is the number itself.

B) False. 1 is a factor of every number, not a multiple (unless the number itself is 1).

C) False. A number has an infinite number of multiples (e.g., multiples of 3 are 3, 6, 9, 12, ... and so on indefinitely).

D) True. Multiples of 5 are 5, 10, 15, 20, 25, etc. All these numbers indeed end in either 0 or 5.

Solution:

The smallest prime number is 2, as it is the smallest number greater than 1 with exactly two factors (1 and 2).

To find the smallest composite number, we check numbers greater than 1: 2 is prime, 3 is prime. The next number is 4, which has factors 1, 2, and 4. So, 4 is the smallest composite number.

The sum of the smallest prime number and the smallest composite number is 2 + 4.

2 + 4 = 6.