For all integers a, b, and c, if the expression a^3 b - b^3 c + c^3 a is an odd integer, t… — GMAT

For all integers $a$ , $b$ , and $c$ , if the expression $a^3 b - b^3 c + c^3 a$ is an odd integer, then the product $(a - b)(b - c)(c - a)$ must be divisible by 4.

Cevap: Cevap

Cevap

The statement is False.

The statement claims that the product must be divisible by 4 for all integer inputs where the given expression is odd. However, setting two variables to odd integers and one variable to an even integer (such as

a=3, b=1, c=0

) makes the initial expression odd (

27

) while producing a product of

-6

, which is not divisible by 4.

Adım Adım Çözüm

Simplify the expression modulo 2 to establish necessary parity conditions.

Since

x^3 \equiv x \pmod 2

for any integer

x

, the parity of

x^3 y

is identical to the parity of

xy

. Thus,

a^3b - b^3c + c^3a \equiv ab - bc + ca \pmod 2

Reducing powers modulo 2 determines which parity combinations of

a, b,

and

c

yield an odd result.

Analyze all possible parity combinations for the three variables.

Case 1: All three are odd

\implies odd - odd + odd = odd

.
Case 2: Two are odd and one is even (e.g.,

a, b

odd,

c

even)

\implies odd - even + even = odd

.
Case 3: One is odd and two are even

\implies even - even + even = even

.
Case 4: All three are even

\implies even - even + even = even

Identifying that both Case 1 and Case 2 produce an odd value is essential for finding edge cases.

Test Case 2 using specific integer values to check whether

(a - b)(b - c)(c - a)

must be divisible by 4.

Set

a = 3

(odd),

b = 1

(odd), and

c = 0

(even). The given expression equals

3^3(1) - 1^3(0) + 0^3(3) = 27

, which is odd. Evaluating the product gives

(3 - 1)(1 - 0)(0 - 3) = (2)(1)(-3) = -6

Testing specific valid integers disproves a universal 'must be' claim.

Evaluate the divisibility of the resulting product by 4.

-6

is not divisible by 4 because

-6 / 4 = -1.5

, which is not an integer.

Finding a valid case where the product is not divisible by 4 proves the statement is false.

Anahtar Kavram

Parity rules for algebraic expressions and edge-case evaluation using zero as an even integer

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