When the positive integer n is divided by 5, the remainder is 2. Which of the following must be true?

Solution:

Given that when the positive integer n is divided by 5, the remainder is 2, we can express n as:

n = 5k + 2 where k is an integer.

Option ক)

n is odd

Since n = 5k + 2 the value of n depends on k. If k is even, n is even; if k is odd, n is odd.

Therefore, n is not necessarily odd.

Option খ)

n is even

Similarly, as discussed above, n = 5k + 2 could be odd or even depending on the value of k. If k is even, n is even, but if k is odd, n is odd.

Therefore, n is not necessarily even.

Option গ)

n - 1 is divisible by 3

Let's calculate n - 1:

n - 1 = 5k + 2 - 1 = 5k + 1

We need to check if 5k + 1 is divisible by 3.

We know that any integer 5k leaves a remainder of 2 when divided by 3

∴ For n - 1 to be divisible by 3, 5k + 1 mod  3 must equal 0. This can occur depending on k, but it's not necessarily true for all k.

None of the provided options must be true in all cases. Therefore, none of the options is necessarily true based on the information given.