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There are two numbers p and q such that their HCF is 1. Which of the following statements are correct?
1. Both p and q may be prime.
2. One number may be prime and the other composite.
3. Both the numbers may be composite.
1. Both p and q may be prime.
2. One number may be prime and the other composite.
3. Both the numbers may be composite.
Select the correct answer using the code given below:
- a)1 and 2 only
- b)2 and 3 only
- c)1 and 3 only
- d)1, 2 and 3
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
There are two numbers p and q such that their HCF is 1. Which of the f...
HCF of p and q is 1
1. Both p and q may be prime
Suppose p = 2 and q = 3
⇒ HCF of p and q = 1
⇒ Statement 1 is correct
1. Both p and q may be prime
Suppose p = 2 and q = 3
⇒ HCF of p and q = 1
⇒ Statement 1 is correct
2. One number may be prime, and the other is composite.
Suppose p = 3 and q = 4
⇒ HCF of p and q = 1
⇒ Statement 2 is correct
Suppose p = 3 and q = 4
⇒ HCF of p and q = 1
⇒ Statement 2 is correct
3. Both the numbers may be composite
Suppose p = 9 and q = 10
⇒ HCF of p and q = 1
⇒ Statement 3 is correct.
∴ All three statements are correct.
Suppose p = 9 and q = 10
⇒ HCF of p and q = 1
⇒ Statement 3 is correct.
∴ All three statements are correct.
Most Upvoted Answer
There are two numbers p and q such that their HCF is 1. Which of the f...
Explanation:
To understand this question and its options, let's first revise what the HCF (Highest Common Factor) is. The HCF of two numbers is the largest number that divides both of them without leaving a remainder. If the HCF of two numbers is 1, it means that the two numbers are co-prime or relatively prime.
Now let's analyze each statement and determine whether it is correct or not.
1. Both p and q may be prime:
This statement is correct. If both p and q are prime numbers, their only common factor will be 1. For example, if p = 3 and q = 5, both numbers are prime and their HCF is 1.
2. One number may be prime and the other composite:
This statement is correct as well. If one number is a prime number and the other is a composite number, their HCF can still be 1. For example, if p = 2 (prime) and q = 6 (composite), the only factor they have in common is 1.
3. Both the numbers may be composite:
This statement is also correct. If both p and q are composite numbers, they can still have an HCF of 1. For example, if p = 4 and q = 9, both numbers are composite but their only common factor is 1.
Therefore, all three statements are correct, and the correct answer is option D: 1, 2, and 3.
In summary:
- Both p and q may be prime.
- One number may be prime and the other composite.
- Both numbers may be composite.
All of these scenarios can have an HCF of 1, indicating that the numbers are co-prime or relatively prime.
To understand this question and its options, let's first revise what the HCF (Highest Common Factor) is. The HCF of two numbers is the largest number that divides both of them without leaving a remainder. If the HCF of two numbers is 1, it means that the two numbers are co-prime or relatively prime.
Now let's analyze each statement and determine whether it is correct or not.
1. Both p and q may be prime:
This statement is correct. If both p and q are prime numbers, their only common factor will be 1. For example, if p = 3 and q = 5, both numbers are prime and their HCF is 1.
2. One number may be prime and the other composite:
This statement is correct as well. If one number is a prime number and the other is a composite number, their HCF can still be 1. For example, if p = 2 (prime) and q = 6 (composite), the only factor they have in common is 1.
3. Both the numbers may be composite:
This statement is also correct. If both p and q are composite numbers, they can still have an HCF of 1. For example, if p = 4 and q = 9, both numbers are composite but their only common factor is 1.
Therefore, all three statements are correct, and the correct answer is option D: 1, 2, and 3.
In summary:
- Both p and q may be prime.
- One number may be prime and the other composite.
- Both numbers may be composite.
All of these scenarios can have an HCF of 1, indicating that the numbers are co-prime or relatively prime.
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