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State 'T' for true and 'F' for false. ...

State 'T' for true and 'F' for false.
(i) If an even number is divided by 2, then the quotient is always odd.
(ii) All even numbers are composite numbers.
(iii)The L.C.M. of two co-prime numbers cannot be equal to their product.
(iv) Every number is a factor of itself.

A

`{:("(i) (ii) (iii) (iv)"),("T T F F"):}`

B

`{:("(i) (ii) (iii) (iv)"),("F F F T"):}`

C

`{:("(i) (ii) (iii) (iv)"),("T F T T"):}`

D

`{:("(i) (ii) (iii) (iv)"),("F F T T"):}`

Text Solution

AI Generated Solution

The correct Answer is:
Let's analyze each statement one by one to determine if they are true (T) or false (F). ### Step-by-Step Solution: 1. **Statement (i): If an even number is divided by 2, then the quotient is always odd.** - **Analysis**: An even number can be expressed as \(2k\), where \(k\) is any integer. When we divide \(2k\) by 2, we get: \[ \frac{2k}{2} = k \] Here, \(k\) can be either even or odd depending on the value of \(k\). For example: - If \(k = 3\) (odd), then \(2k = 6\) and \(\frac{6}{2} = 3\) (odd). - If \(k = 4\) (even), then \(2k = 8\) and \(\frac{8}{2} = 4\) (even). - **Conclusion**: The quotient can be either odd or even. Therefore, this statement is **False (F)**. 2. **Statement (ii): All even numbers are composite numbers.** - **Analysis**: A composite number is defined as a number that has more than two distinct positive divisors. The number 2 is an even number but it only has two divisors: 1 and 2. Therefore, it is a prime number, not composite. - **Conclusion**: Since not all even numbers are composite (specifically, 2 is not), this statement is **False (F)**. 3. **Statement (iii): The L.C.M. of two co-prime numbers cannot be equal to their product.** - **Analysis**: Co-prime numbers are numbers that have no common factors other than 1. For two co-prime numbers \(a\) and \(b\), the relationship between LCM and GCD is given by: \[ \text{LCM}(a, b) \times \text{GCD}(a, b) = a \times b \] Since the GCD of co-prime numbers is 1, we have: \[ \text{LCM}(a, b) \times 1 = a \times b \implies \text{LCM}(a, b) = a \times b \] - **Conclusion**: The LCM of two co-prime numbers is indeed equal to their product. Therefore, this statement is **False (F)**. 4. **Statement (iv): Every number is a factor of itself.** - **Analysis**: By definition, a factor of a number is a whole number that can be divided evenly into that number. Every number \(n\) can be divided by itself, resulting in 1. Thus, \(n\) is a factor of \(n\). - **Conclusion**: This statement is **True (T)**. ### Final Answers: - (i) F - (ii) F - (iii) F - (iv) T
Let's analyze each statement one by one to determine if they are true (T) or false (F). ### Step-by-Step Solution: 1. **Statement (i): If an even number is divided by 2, then the quotient is always odd.** - **Analysis**: An even number can be expressed as \(2k\), where \(k\) is any integer. When we divide \(2k\) by 2, we get: \[ \frac{2k}{2} = k ...
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