If R and S are different integers both divisible by 5, then which of the following is not necessarily true ?

The correct option is R + S is divisible by 10.

Let us analyze why this statement is not necessarily true, while all the other options must be true, given that $R$ and $S$ are different integers both divisible by 5.

Since $R$ and $S$ are divisible by 5, we can write them as:
$R=5x$
$S=5y$
where $x$ and $y$ are distinct integers (since $R\ne S$).

Let us evaluate each option to see if it is always true or not:

1. Evaluating R - S is divisible by 5:
We have:
$R-S=5x-5y=5(x-y)$
Since $x-y$ is an integer, $R-S$ is a multiple of 5. Therefore, this statement is always true.

2. Evaluating R + S is divisible by 10:
We have:
$R+S=5x+5y=5(x+y)$
For $R+S$ to be divisible by 10, the term $x+y$ must be an even number. However, this is not always the case. For example, if we choose $R=5$ (so $x=1$) and $S=10$ (so $y=2$), both are divisible by 5. Then:
$R+S=5+10=15$
The sum 15 is not divisible by 10. Thus, this statement is not necessarily true.

3. Evaluating R × S is divisible by 25:
We have:
$R\times S=\left(5x\right)\times \left(5y\right)=25xy$
Since $xy$ is an integer, the product is a multiple of 25. Therefore, this statement is always true.

4. Evaluating R² + S² is divisible by 5:
We have:
$R^2+S^2=(5x{)}^2+(5y{)}^2=25x^2+25y^2=5(5x^2+5y^2)$
Since the expression inside the parenthesis is an integer, the sum of squares is a multiple of 5 (in fact, it is also divisible by 25). Thus, this statement is always true.