If $(126)_n = 86_{10}$, find the positive value of $n$.

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01-11-2026
Mathematics

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If $(126)_n = 86_{10}$, find the positive value of $n$.

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-01T15:53:05-05:00

Translate the base-$n$ representation $(126)_n$ into the algebraic equation $n^2 + 2n + 6 = 86$.
Simplify the equation to the quadratic form $n^2 + 2n - 80 = 0$.
Factor the equation to find the roots, resulting in $n = -10$ and $n = 8$.
Apply the base validity constraint $n > 6$ to select the only valid positive integer solution, $\boxed{8}$.

Explanation

Analysis and Base Representation Conditions

The number given in base $n$ is $(126)_n$, which is equivalent to the decimal number $86_{10}$. A number represented in base $n$ with digits $d_2d_1d_0$ is calculated by the formula $d_2 \cdot n^2 + d_1 \cdot n^1 + d_0 \cdot n^0$. Furthermore, a fundamental rule of base representations is that each digit must be strictly less than the base itself. In our case, the digits are 1, 2, and 6, so we must have the condition $n > 6$.

Setting up the Quadratic Equation

We express the given base-$n$ number as a polynomial in $n$:
\[1 \cdot n^2 + 2 \cdot n + 6 = 86\]
Subtracting 86 from both sides to form a standard quadratic equation:
\[n^2 + 2n + 6 - 86 = 0\]
\[n^2 + 2n - 80 = 0\]

Solving the Quadratic Equation

We can solve the quadratic equation $n^2 + 2n - 80 = 0$ by factoring. We look for two numbers that multiply to $-80$ and add to $2$. These numbers are $10$ and $-8$:
\[(n + 10)(n - 8) = 0\]
This gives us two potential solutions: $n = -10$ and $n = 8$.

Verifying the Condition and Final Answer

Recall our condition from the first step: because the digit 6 appears in the representation $(126)_n$, the base must satisfy $n > 6$. Comparing our two potential solutions, $-10$ and $8$, we see that only $8$ satisfies this condition ($8 > 6$). Therefore, the valid base is $n = 8$.

Examples

Base representation is foundational in computer science, where data is often represented in binary (base 2), octal (base 8), or hexadecimal (base 16). For example, if a computer system uses a base-8 numbering system for file permissions, knowing how to convert between these systems and decimal helps in understanding how those permissions are calculated and stored. This exercise demonstrates how to determine the underlying base when the representation is known, a common task when deciphering encoded or legacy data systems.