For all non-zero real numbers a and b, the binary operator \diamond is defined by a \diamo… — GMAT

For all non-zero real numbers $a$ and $b$ , the binary operator $\diamond$ is defined by $a \diamond b = a - \frac{b}{a}$ . If $k$ is a positive integer such that $(k \diamond 3) \diamond 3 = \frac{1}{2}$ , what is the value of $k$ ?

Answer: 3

Answer

The positive integer value of

k

is 3.

Applying the non-commutative custom operator definition

a \diamond b = a - \frac{b}{a}

to the nested expression

(k \diamond 3) \diamond 3 = \frac{1}{2}

requires working from the inside out. Letting

u = k \diamond 3 = k - \frac{3}{k}

, the outer equation becomes

u \diamond 3 = u - \frac{3}{u} = \frac{1}{2}

. Multiplying through by

2u

gives

2u^2 - u - 6 = 0

, which factors as

(2u + 3)(u - 2) = 0

, yielding

u = 2

u = -\frac{3}{2}

. Substituting

u = 2

back into

k - \frac{3}{k} = 2

produces

k^2 - 2k - 3 = 0

, which factors as

(k - 3)(k + 1) = 0

. Since

k

must be a positive integer,

k = 3

. Testing

u = -\frac{3}{2}

gives

2k^2 + 3k - 6 = 0

, which has no integer solutions.

Step-by-Step Solution

Substitute a temporary variable for the inner custom operator expression.

Let

u = k \diamond 3 = k - \frac{3}{k}

Simplifies the nested expression

(k \diamond 3) \diamond 3 = \frac{1}{2}

to a single variable operation

u \diamond 3 = \frac{1}{2}

Apply the operator definition to

u \diamond 3

and solve the resulting quadratic equation for

u

u - \frac{3}{u} = \frac{1}{2} \implies 2u^2 - u - 6 = 0 \implies (2u + 3)(u - 2) = 0

, so

u = 2

u = -\frac{3}{2}

Clearing the denominator

u

transforms the custom operator equation into a standard quadratic equation.

Solve for positive integer

k

using the obtained values of

u

Setting

k - \frac{3}{k} = 2

yields

k^2 - 2k - 3 = 0 \implies (k - 3)(k + 1) = 0

. Since

k

must be a positive integer,

k = 3

. Setting

k - \frac{3}{k} = -\frac{3}{2}

yields

2k^2 + 3k - 6 = 0

, which has non-integer discriminant

\sqrt{57}

Only

u = 2

yields an integer solution for

k

, satisfying all problem constraints.

Key Concept

Custom Operator Evaluation and Nested Functional Equations

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