If $M$ and $T$ are odd numbers, and $M$ is a multiple of $T$ , which of the following must be true? $\newline$ A. $M + T$ is odd. $\newline$ B. $MT$ is even. $\newline$ C. $M - T$ is odd. $\newline$ D. $M \div T$ is odd.

Full solution

Q. If

M

and

T

are odd numbers, and

M

is a multiple of

T

, which of the following must be true?

\newline

M + T

is odd.

\newline

MT

is even.

\newline

M - T

is odd.

\newline

M \div T

is odd.

Odd Numbers Property: Let's consider the properties of odd numbers. When we add two odd numbers, the sum is always even. This is because an odd number can be expressed as $2n + 1$ , where $n$ is an integer. Adding two such expressions $(2n + 1) + (2m + 1)$ results in $2(n + m + 1)$ , which is an even number. Let's apply this to option $A$ .
Evaluate $M + T$ : Calculate $M + T$ using the property of odd numbers. If $M$ is odd and $T$ is odd, then $M + T$ must be even. Therefore, option $A$ , which states that $M + T$ is odd, is incorrect.
Evaluate MT: Now let's consider option B, which states that MT is even. The product of two odd numbers is always odd. This is because the product of $(2n + 1)$ and $(2m + 1)$ is $4nm + 2n + 2m + 1$ , which is odd. Therefore, option B is incorrect.
Evaluate $M - T$ : Next, let's evaluate option C, which states that $M - T$ is odd. Since $M$ and $T$ are both odd, their difference $M - T$ will be even, similar to the sum of two odd numbers. Therefore, option C is incorrect.
Evaluate $M \div T$ : Finally, let's consider option D, which states that $M \div T$ is odd. Since $M$ is a multiple of $T$ and both are odd, their quotient $M \div T$ must be an odd number. This is because the only way for an odd number to be a multiple of another odd number is for the multiplier itself to be odd. Therefore, option D must be true.