Question

If \(-\dfrac{m}{19}\) is an even integer, which of the following must be true?

• A. \( m \) is a negative number.
• B. \( m \) is a positive number.
• C. \( m \) is a prime number.
• D. \( m \) is an odd integer.
• E. \( m \) is an even integer.

Hint:

When dealing with integer divisibility conditions, always rewrite the expression in terms of a multiple of an integer. This quickly reveals properties like evenness or oddness.

Answer 1

Answer: (E)

Step 1: Condition analysis.
We are told that \(-\dfrac{m}{19}\) is an even integer. Let \(-\dfrac{m}{19} = 2k\), where \( k \) is an integer.
Step 2: Express \( m \).
\[ -\frac{m}{19} = 2k \quad \Rightarrow \quad m = -19(2k) = -38k \] Thus, \( m \) is always a multiple of 38.
Step 3: Implication.
Since 38 is an even number, any multiple of 38 must also be even. Hence, \( m \) must be an even integer.
Step 4: Elimination of other options.
- (A) \( m \) does not have to be negative. Example: if \( m = 38 \), then \(-\dfrac{38}{19} = -2\), which is even.
- (B) \( m \) does not have to be positive, since \( m = -38 \) also works.
- (C) \( m \) is not necessarily prime, as multiples of 38 are not prime.
- (D) \( m \) is not odd, since it must be a multiple of 38.
Step 5: Conclusion.
The only guaranteed truth is that \( m \) is an even integer. \[ \boxed{\text{(E) \( m \) is an even integer.}} \]