If n Is a Positive Integer And r Is The Remainder When n^2 - 1 is Divided By 8 GMAT Data Sufficiency

Question: If n is a positive integer and r is the remainder when n^2 - 1 is divided by 8, what is the value of r?

(1) n is odd
(2) n is not divisible by 8

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are not sufficient.

‘If n is a positive integer and r is the remainder when n^2 - 1 is divided by 8, what is the value of r?’ - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Official Guide 2022". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Solution and Explanation:

Approach Solution 1:
Given: n is a positive integer and r is the remainder when n^2 - 1 is divided by 8
Find Out: what is the value of r?
Let us consider the equation:
n^2-1=(n-1)(n+1)
Now we will check both the options provided:
Statement 1) n is odd --> both n-1 and n+1 are even.
Moreover, they are consecutive even integers thus one of them is divisible by 4 too.
Now, as one is divisible by 2 and another by 4 then (n-1)(n+1) is divisible by 2*4=8.
Hence, the statement is Sufficient.
Statement 2) n is not divisible by 8 --> try n=1 to get an YES answer and n=2 to get a NO answer.
Hence, the statement is Not sufficient.
The correct answer is A.

Correct Answer: A

Approach Solution 2:
Given:

• n is positive interger and
• n^2 - 1 = 8 * k + r -> r remainder

Find Out:

• what is the value of r??

Statement 1) n is odd
n^2-1 = (n+1) * (n-1)
so n+1 and n-1 are consequetive even numbers.
One of them will be multiple of 2 and the other will be multiple of 4.
So n^2 - 1 will be evenly divided by 8 and r=0
Hence, Sufficient
Statement 2) n is not divisible by 8.
Hence, Not sufficient.
Since statement 1 is sufficient alone, A is the correct answer.

Correct Answer: A

Approach Solution 3:

Case (1) n is odd
n = 2m + 1
this implies n^2 - 1 = (2m+1)^2 - 1 = 4m^2 + 4m = 4m(m+1)
The remainder when n^2 - 1 is divided by 8 = 0 = r
SUFFICIENT

Case (2) n is not divisible by 8
this implies if n= 5 then n^2 - 1 = 24; r = 0
this implies if n=6 then n^2 - 1 = 35; r = 3
NOT SUFFICIENT

Correct Answer: A

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