Steps 1 & 2: Understand Question and Draw Inferences
- Difference of two integers will be odd, only if one of the numbers is odd.
- Let’s see when R and S will be odd:
- The even-odd nature of R will depend on the values of n and n + 1. Since n and n+ 1 are consecutive integers, only one of them can be even.
- Now, in the expression of R as n(n+1) is divided by 2, for R to be even, either of n or n + 1 should be a multiple of 4, i.e. they should be of the form 4k, where k is a positive integer. Hence, following cases arise:
- n is of the form 4k, i.e. n = 4k OR
- n+1 is of the form 4k, i.e. n + 1 = 4k
- So, n = 4k – 1 = 4(k -1) + 4 -3 = 4(k-1) + 3
- So, we can say that R will be even if n is of the form 4k or 4k + 3. For all the other cases, R will be odd
- Since the expression of S is similar to R, we can say that S will be even if m is of the form 4k or 4k + 3
- So, we need to look for values of m and n in the statements.
Step 3: Analyze Statement 1 independently
(1) m is odd and n is even
- As we do not know if m or n is of the form of 4k or 4k + 3, we cannot say, if R and S are odd or even.
- Insufficient to answer.
Step 4: Analyze Statement 2 independently
(2) m can be expressed in the form of 4x +3 and n can be expressed in the form of 2x, where x is a positive integer
- As m can be expressed in the form of 4x + 3, S will be even.
- As n can be expressed in the form of 2x, n may be expressed in the form of 4x or may not be expressed in the form of 4x.
- So, R may be odd or R may be even.
- If R is odd, R – S = odd
- If R is even, R – S = even.
- Hence, we cannot say for sure if R – S is odd.
- Insufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
(1) From statement-1, we know that is odd and n is even
(2) From statement-2, we know that R may be even or odd and S is even
- Combining both the statements does not tell us anything about the even-odd nature of R, hence we cannot say for sure if R – S = odd.
- Insufficient to answer.
Answer: E