All questions of Prime Time for Class 6 Exam

A number divisible by both 2 and 3 must be even and a multiple of 3. 12 fits this description.

Understanding Prime Numbers
Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. In simpler terms, a prime number can only be divided evenly by 1 and the number itself.
Identifying the Options
Let's analyze the given options:

• 8: The factors of 8 are 1, 2, 4, and 8. Since it has divisors other than 1 and 8, it is not a prime number.
• 9: The factors of 9 are 1, 3, and 9. It also has divisors other than itself and 1, so 9 is not a prime number.
• 11: The factors of 11 are only 1 and 11. Since it has no other divisors, 11 is a prime number.
• 12: The factors of 12 are 1, 2, 3, 4, 6, and 12. Due to multiple divisors, 12 is not a prime number.

Conclusion
From the analysis above, the only number that meets the criteria of a prime number is:

• Option C: 11

Thus, the correct answer is option 'C' as it is the only prime number among the options listed. Understanding prime numbers helps in various areas of mathematics, including number theory and cryptography.

Co-prime numbers have no common factors other than 1. 9 and 28 are co-prime because they have no common factors other than 1.

Understanding Multiples of 4
To determine which numbers are multiples of 4, we need to understand what a multiple is. A number is a multiple of 4 if it can be divided by 4 without leaving a remainder.
Checking the Options
Let's evaluate each of the given options:
- Option a: 14
- When we divide 14 by 4, we get 3 with a remainder of 2.
- 14 is not a multiple of 4.
- Option b: 18
- Dividing 18 by 4 gives us 4 with a remainder of 2.
- Thus, 18 is not a multiple of 4.
- Option c: 20
- When we divide 20 by 4, we get exactly 5 with no remainder.
- This confirms that 20 is a multiple of 4.
- Option d: 22
- Dividing 22 by 4 results in 5 with a remainder of 2.
- Hence, 22 is not a multiple of 4.
Conclusion
Among the options provided, the only number that divides evenly by 4 is 20. Therefore, the correct answer is:
Correct Answer: Option C - 20

Common Factor of 7 and 9

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Explanation:

• Factors: Factors are numbers that can be multiplied together to get another number. For example, factors of 7 are 1 and 7 (since 1 x 7 = 7), and factors of 9 are 1, 3, and 9 (since 1 x 9 = 9 and 3 x 3 = 9).
• Common Factor: Common factors are factors that two or more numbers have in common. In this case, the numbers 7 and 9 only have one common factor.
• Common Factor of 7 and 9: The only common factor of 7 and 9 is 1. This is because 1 is a factor of every number, and in this case, it is the only factor that both 7 and 9 share.

Therefore, the only common factor of 7 and 9 is 1.

Understanding Prime Factorization
Prime factorization involves breaking down a composite number into its prime factors. Prime numbers are those that have exactly two distinct positive divisors: 1 and the number itself. For example, the prime numbers are 2, 3, 5, 7, 11, etc.
Finding the Prime Factorization of 12
To find the prime factorization of 12, we can use the following steps:
1. Divide by the Smallest Prime
- Start with the smallest prime number, which is 2.
- Divide 12 by 2:
12 ÷ 2 = 6
- So, we have one factor as 2.
2. Continue Factoring
- Now, take the result (6) and divide by 2 again:
6 ÷ 2 = 3
- We now have another factor as 2.
3. Final Factorization
- Now we are left with 3, which is a prime number itself, so we stop here.
- Thus, the complete prime factorization of 12 is:
2 × 2 × 3
Conclusion
Hence, the prime factorization of 12 is correctly represented by option 'A':
- 2 × 2 × 3
This shows that the number 12 can be expressed as the product of its prime factors.
Other Options Explained
- Option B (3 × 4): 4 is not a prime number (it factors to 2 × 2).
- Option C (2 × 6): 6 is also not prime (it factors to 2 × 3).
- Option D (1 × 12): 1 is not considered a prime factor.
This confirms that option A is indeed the correct answer!

Understanding Distinct Prime Numbers
Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. A number that is the product of exactly three distinct prime numbers can be represented as p1 * p2 * p3, where p1, p2, and p3 are distinct prime numbers.
Analysis of Each Option
- Option a: 20
- Prime factorization: 20 = 2 * 2 * 5
- Distinct primes: 2, 5
- Count: Only 2 distinct primes.
- Option b: 165
- Prime factorization: 165 = 3 * 5 * 11
- Distinct primes: 3, 5, 11
- Count: Exactly 3 distinct primes.
- Option c: 45
- Prime factorization: 45 = 3 * 3 * 5
- Distinct primes: 3, 5
- Count: Only 2 distinct primes.
- Option d: 147
- Prime factorization: 147 = 3 * 7 * 7
- Distinct primes: 3, 7
- Count: Only 2 distinct primes.
Conclusion
Among the options provided, only option b (165) is the product of exactly three distinct prime numbers: 3, 5, and 11. Thus, the correct answer is option 'B'.

Understanding Factors
Factors are the whole numbers that can be multiplied together to produce another number. In this case, we need to find which of the given options can evenly divide 24 without leaving a remainder.
Identifying Factors of 24
To determine the factors of 24, we can list the pairs of numbers that multiply to give 24:

• 1 x 24 = 24
• 2 x 12 = 24
• 3 x 8 = 24
• 4 x 6 = 24

From this list, we see that the factors of 24 are:

• 1
• 2
• 3
• 4
• 6
• 8
• 12
• 24

Evaluating the Options
Now, let’s evaluate the given options:

• Option A: 3 - Yes, 3 is a factor of 24 (8 x 3 = 24).
• Option B: 5 - No, 5 is not a factor (24 ÷ 5 = 4.8).
• Option C: 7 - No, 7 is not a factor (24 ÷ 7 = 3.4286).
• Option D: 9 - No, 9 is not a factor (24 ÷ 9 = 2.6667).

Conclusion
Among the options provided, the correct answer is:
Option A: 3
This option is the only number that divides 24 evenly, confirming it as a factor of 24.

Understanding Composite Numbers
Composite numbers are defined as numbers that have more than two distinct positive divisors. In other words, they can be divided evenly by numbers other than 1 and themselves.
Identifying Composite Numbers from the Options
Let’s analyze the given options:

• 5: This number has only two divisors, which are 1 and 5. Therefore, it is a prime number.
• 7: Similar to 5, 7 also has only two divisors: 1 and 7. Thus, it is a prime number.
• 11: This number, too, has just two divisors: 1 and 11. Hence, it is a prime number.
• 12: This number has several divisors: 1, 2, 3, 4, 6, and 12. Since it can be divided evenly by numbers other than 1 and itself, it is classified as a composite number.

Conclusion
Among the options provided, 12 stands out as the only composite number. The other numbers (5, 7, and 11) are all prime, with only two divisors each. Knowing how to categorize numbers as composite or prime is essential in mathematics, especially at this level of study.

2 is the only number that can divide both 14 and 36 evenly.

Understanding Co-prime Numbers
Co-prime numbers, also known as relatively prime numbers, are pairs of numbers that have no common factors other than 1. This means their greatest common divisor (GCD) is 1.
Analyzing the Options
Let’s evaluate each pair of numbers to determine if they are co-prime:
Option A: 4 and 9
- Factors of 4: 1, 2, 4
- Factors of 9: 1, 3, 9
- Common factors: 1
Since the only common factor is 1, 4 and 9 are co-prime.
Option B: 12 and 15
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 15: 1, 3, 5, 15
- Common factors: 1, 3
Since they share a common factor (3), they are not co-prime.
Option C: 18 and 24
- Factors of 18: 1, 2, 3, 6, 9, 18
- Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
- Common factors: 1, 2, 3, 6
Since they share multiple common factors, they are not co-prime.
Option D: 20 and 30
- Factors of 20: 1, 2, 4, 5, 10, 20
- Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
- Common factors: 1, 2, 5, 10
Again, since they share multiple common factors, they are not co-prime.
Conclusion
Hence, the only pair that qualifies as co-prime is Option A: 4 and 9. They have no common factors other than 1, making them relatively prime to each other.

The least common multiple of 3 and 5 is the smallest number that both 3 and 5 divide into, which is 15.

The smallest prime number is 2, as it can only be divided by 1 and 2.

A multiple of 5 ends in 0 or 5. 32 does not end in 0 or 5, so it is not a multiple of 5.