If 3(m+n) is even, then which of the following must also be even?
A. m+n
B. m
C. n
D. 3m
E. 3n+1
thanks
A, m+n. multiplying any odd number by three results in another odd number, so m+n must be even
To determine which of the options is necessarily even when 3(m+n) is even, let's analyze each option:
A. m+n: Since we are given that 3(m+n) is even, we cannot determine whether m+n is even or odd. This option is not necessarily even.
B. m: We cannot determine from the given information whether m is even or odd. Therefore, this option is not necessarily even.
C. n: Similar to option B, we cannot determine from the given information whether n is even or odd. This option is not necessarily even.
D. 3m: Since 3(m+n) is even, we can say that 3m is even, as it is a multiple of 3. This option is necessarily even.
E. 3n+1: This term is not necessarily even. If n is even, the entire term will be odd since 3 multiplied by an even number is even, and adding 1 to an even number results in an odd number. Therefore, this option is not necessarily even.
Therefore, the correct answer is D. 3m, as it is the only option that must be even when 3(m+n) is even.
To determine which of the given expressions must also be even when 3(m+n) is even, let's analyze the factors involved.
Starting with 3(m+n), we know that if it is even, it must be divisible by 2. Since 3 is not divisible by 2, it follows that (m+n) must be divisible by 2 in order for the product 3(m+n) to be even.
Now let's evaluate each option provided:
A. m+n: As explained above, if (m+n) is divisible by 2, m+n itself will be even. Therefore, A is a possible answer.
B. m: In order for m to be even, it must be divisible by 2. However, we cannot determine if m is divisible by 2 based solely on the fact that 3(m+n) is even. Therefore, B is not a must-be-even option.
C. n: Similar to m, we cannot determine whether n is divisible by 2 from the given information. Thus, C is not a must-be-even option.
D. 3m: Since 3 is odd, if 3m were to be even, m would need to be divisible by 2. Nevertheless, this cannot be deduced from the given information, so D is not a must-be-even option.
E. 3n+1: For 3n+1 to be even, 3n would need to be odd, and that would require n to be odd. However, we don't know whether n is odd or even, so E is also not a must-be-even option.
To recap, the only expression that must be even when 3(m+n) is even is A. m+n.