If $R$ and $S$ are different integers both divisible by $5$, then which of the following is not necessarily true?
A. $\left( {R - S} \right)$ is divisible by $5$
B. $\left( {R + S} \right)$ is divisible by $5$
C. $\left( {{R^2} + {S^2}} \right)$ is divisible by $5$
D. $\left( {R + S} \right)$ is divisible by $10$

Hint: Here we are asked to find which of the given following is not true. For that we first assume two numbers which are multiples of $5$ in general form as it is given that they are divisible by five. Then we will form the given all expressions and simplify them to see whether they satisfy its respective conditions.

Complete step-by-step answer:
It is given that $R$ and $S$ are different integers both divisible by $5$, we aim to find which is the false statement from the given options.
To solve this problem let us first take two numbers which are multiples of five since it is given that they are divisible by five.

Let the integers $R$ and $S$ be $5a$ and $5b$ where $a$ and $b$ are integers. Now to check whether the first option is true or not we first need to form the integers in the form $\left( {R - S} \right)$.
$\left( {R - S} \right) = 5a - 5b = 5\left( {a - b} \right)$
Since $5\left( {a - b} \right)$ is a multiple of five it is divisible by five. Thus, the first option is true.

Let’s check whether the second option is true or not. We first need to form the integers in the form $\left( {R + S} \right)$.
$\left( {R + S} \right) = 5a + 5b = 5\left( {a + b} \right)$
Since $5\left( {a + b} \right)$ is a multiple of five it is divisible by five. Thus, the second option is true.

Now to check whether the third option is true or not we first need to form the integers in the form $\left( {{R^2} + {S^2}} \right)$.
$\left( {{R^2} + {S^2}} \right) = 25{a^2} + 25{b^2} = 25\left( {{a^2} + {b^2}} \right) = 5 \times 5\left( {{a^2} + {b^2}} \right)$
Since $5 \times 5\left( {{a^2} + {b^2}} \right)$ is a multiple of five it is divisible by five. Thus, the third option is true.

Let us check the last option now for that we need to form $\left( {R + S} \right)$ we already have simplified this form for the second option let us use that
$\left( {R + S} \right) = 5a + 5b = 5\left( {a + b} \right)$
We know that this is divisible by five since this is a multiple of five but this is not divisible by ten. Thus, the last option is false.
That is option d) $\left( {R + S} \right)$ is divisible by $10$ is not true.

So, the correct answer is “Option A, B and C”.

Note: The above type of problems can only be solved generally since they have been given generally for all integers. If we want to check our answer, we can take any two integers to see whether we get the same answer. For example: let $a = 4$ and $b = - 2$ then $\left( {R + S} \right) = 5\left( 4 \right) + 5\left( { - 3} \right) = 5\left( {4 - 3} \right) = 5$ we know five is not divisible by ten thus we can confirm that our answer is correct. The statement $\left( {R + S} \right)$ is divisible by $10$ is not applicable for all integers.