If m and n are integers, which of the following must be an even integer?
A. 2mn
B. mn
C. mn+2
d. mn-2
e. 3mn
D
Well, well, well. Let's have a look at this question, shall we? Now, remember that an even integer is divisible by 2. So, which of these options will definitely give us an even result?
A. We have 2mn. Well, if either m or n is even, then we're good to go. But if both m and n are odd, then we're out of luck.
B. Now we have mn. Well, this could be even or odd depending on the values of m and n. It's not guaranteed to be even, my friend.
C. Ah-ha! Now we're talking. With mn+2, no matter what the values of m and n are, we're adding 2 to the result. And if we remember correctly, 2 is an even number. Bingo! We've found our answer!
D. With mn-2, just like the previous option, it all depends on the values of m and n. Not a guarantee of evenness here.
E. Lastly, we have 3mn. Well, as long as at least one of m or n is even, we're fine. But if both are odd, we're back to square one.
So, my dear question-asker, the only option that must result in an even integer is C. Take that, odd numbers!
To determine which of the given options must be an even integer, we need to consider the properties of even and odd numbers.
An even number is defined as any integer that is divisible by 2, while an odd number is not divisible by 2.
Let's evaluate each option:
A. 2mn
Since 2 is a factor of this expression, it must be an even integer.
B. mn
The product of any two integers can be even or odd depending on the factors. For example, if m = 1 and n = 3, then mn = 1*3 = 3, which is odd. Therefore, this expression may or may not be even.
C. mn+2
If mn is even, then adding 2 will result in an even integer. However, if mn is odd, adding 2 will result in an odd integer. Therefore, this expression may or may not be even.
D. mn-2
Similar to option C, if mn is even, then subtracting 2 will result in an even integer. However, if mn is odd, subtracting 2 will result in an odd integer. Therefore, this expression may or may not be even.
E. 3mn
Since there is a factor of 3 in this expression, it cannot be guaranteed to be an even integer.
In conclusion, option A (2mn) is the only one that must be an even integer.
To determine which of the following options must be an even integer, we need to understand the properties of even and odd integers.
An integer is considered even if it is divisible by 2 without leaving a remainder. Conversely, an integer is considered odd if it is not divisible by 2 and leaves a remainder.
Let's analyze each option:
A. 2mn
To determine if this option is even, we need to check if 2mn is divisible by 2. Since 2 is a factor of 2mn, this expression must be an even integer.
B. mn
The product of two integers (m and n) does not necessarily guarantee the result to be even. For example, if m = 2 and n = 3, mn = 6, which is even. However, if m = 2 and n = 5, mn = 10, which is also even. In either case, mn can be odd or even, so this option is not necessarily an even integer.
C. mn + 2
Adding an even number (2) to any integer (mn) does not change its parity. If mn is even, adding 2 to it will still result in an even number. If mn is odd, adding 2 to it will result in an odd number. Therefore, this option is not guaranteed to be even.
D. mn - 2
Similarly to option C, subtracting an even number (2) from an integer (mn) will not change its parity. Therefore, this option is not guaranteed to be even.
E. 3mn
Multiplying an odd number (3) by any integer (mn) will always result in an odd number. Therefore, this option is not guaranteed to be even.
Based on our analysis, the only option that must be an even integer is:
A. 2mn