If r and s are positive integers, is r^2 - s^2 divisible by 8? (1) r and s are both odd in… — GMAT

If $r$ and $s$ are positive integers, is $r^2 - s^2$ divisible by 8?

(1) $r$ and $s$ are both odd integers.
(2) $r + s$ is divisible by 4.

A
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.Answer
E
Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

Each statement alone is sufficient to answer the question with a definitive 'Yes'.

The correct response identifies that each statement alone gives a definitive 'Yes' answer. Statement (1) shows that the square of any odd integer is

1 \pmod 8

, so subtracting two odd squares gives

0 \pmod 8

. Statement (2) shows

r+s

is a multiple of 4, which forces

r

and

s

to share parity, making

r-s

even and their product

(r+s)(r-s)

a multiple of

4 \times 2 = 8

Step-by-Step Solution

Rephrase the question target using algebraic factorization.

Note that

r^2 - s^2 = (r + s)(r - s)

. The question asks whether

(r + s)(r - s)

is a multiple of 8.

Factoring difference of squares isolates the sum and difference components to easily test parity and divisibility.

Evaluate Statement (1):

r

and

s

are both odd integers.

Any odd integer can be written as

2k + 1

. Squaring gives

(2k + 1)^2 = 4k^2 + 4k + 1 = 4k(k + 1) + 1

. Since

k(k + 1)

is the product of two consecutive integers, it is always even, making

4k(k + 1)

a multiple of 8. Thus, any odd integer squared leaves a remainder of 1 when divided by 8. Therefore,

r^2 \equiv 1 \pmod 8

and

s^2 \equiv 1 \pmod 8

, so

r^2 - s^2 \equiv 1 - 1 = 0 \pmod 8

Statement (1) alone yields a definitive 'Yes', so Statement (1) is sufficient.

Evaluate Statement (2):

r + s

is divisible by 4.

r + s

is divisible by 4, then

r + s = 4m

for some integer

m

. Since

r + s

is even,

r

and

s

must have the same parity (either both even or both odd). Consequently, their difference

r - s = (r + s) - 2s = 4m - 2s = 2(2m - s)

is an even integer (a multiple of 2). Multiplying

(r + s)

(r - s)

yields

(\text{multiple of } 4) \times (\text{multiple of } 2) = \text{multiple of } 8

Statement (2) alone yields a definitive 'Yes', so Statement (2) is sufficient.

Key Concept

Divisibility Rules of Difference of Squares and Parity of Integers

Estimated Time:2m 0s

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