What is the remainder when integer n is divided by two?

Solution:

Pre-analysis:

We are asked for the remainder when n is divided by 2

We know when any integer n is divided by 2, the remainder can be 0 (when n is even) or 1 (when n is odd)

Therefore, the question is technically asking us if integer n is even or odd

Statement 1: When n is divided by 5, the remainder is even

According to this statement, \(n\) can be \(5k, 5k+2, 5k+4, 5k+6\) and so on

The value of n thus can be both even or odd based on the value of \(k\)

Thus, statement 1 alone is not sufficient and we can eliminate options A and D


Statement 2: When n is divided by 7, the remainder is even

Similar to statement 1, the value of \(n\) can be \(7p, 7p+2, 7p+4, 7p+6\) and so on

The value of n thus can be both even or odd based on the value of \(p\)

Thus, statement 2 alone is also not sufficient and we can eliminate option B


Combining:

Let's take 2 cases:

\(n = 35\) (odd) and when n=35 is divided by 5 and 7, the remainder is even i.e., 0

\(n = 70\) (even and when n=35 is divided by 5 and 7, the remainder is even i.e., 0

Thus, even after combining, it is not possible to know if n is even or odd


Hence the right answer is Option E