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

Correct Option (2)

If R and S are different integers both divisible by 5, they can be expressed in the form R = 5k and S = 5m, where k and m are distinct integers. The sum of R and S is R + S = 5k + 5m = 5(k + m). For R + S to be divisible by 10, the term (k + m) must be an even integer. This condition is not always met. For example, if R = 20 (where k = 4) and S = 15 (where m = 3), both R and S are divisible by 5. However, their sum R + S = 20 + 15 = 35, which is not divisible by 10. In this case, k + m = 4 + 3 = 7, an odd integer. Therefore, R + S is not necessarily divisible by 10.

Incorrect Options:

Given that R and S are different integers divisible by 5, they can be represented as R = 5k and S = 5m, where k and m are distinct integers.

Regarding Option 1 (R - S is divisible by 5):

R - S = 5k - 5m = 5(k - m). Since k and m are integers, their difference (k - m) is also an integer. Consequently, 5(k - m) is always a multiple of 5, implying R - S is necessarily divisible by 5.

Regarding Option 3 (R × S is divisible by 25):

R × S = (5k) × (5m) = 25km. Since k and m are integers, their product km is also an integer. Therefore, 25km is always a multiple of 25, implying R × S is necessarily divisible by 25.

Regarding Option 4 (R² + S³³ is divisible by 5):

R² = (5k)² = 25k². Since 25 is divisible by 5, R² is necessarily divisible by 5.

S³³ = (5m)³³ = 5³³m³³. Since 5³³ is divisible by 5, S³³ is necessarily divisible by 5.

As both R² and S³³ are individually divisible by 5, their sum R² + S³³ is also necessarily divisible by 5.