If 132_k + 43_k = 230_k, where k represents a positive integer base, what is the value of … — JAMB UTME

If $132_k + 43_k = 230_k$ , where $k$ represents a positive integer base, what is the value of $k$ ?

The value of the base

k

is 5.

Expanding

132_k

43_k

, and

230_k

into base 10 yields

(k^2 + 3k + 2) + (4k + 3) = 2k^2 + 3k

. Simplifying this gives the quadratic equation

k^2 - 4k - 5 = 0

, which factors as

(k - 5)(k + 1) = 0

. Since a number base must be a positive integer greater than the maximum digit present (which is 4), the correct base is 5.

Expand each term in the equation

132_k + 43_k = 230_k

into base 10 using place-value notation.

132_k = 1 \cdot k^2 + 3 \cdot k^1 + 2 \cdot k^0 = k^2 + 3k + 2

43_k = 4 \cdot k^1 + 3 \cdot k^0 = 4k + 3

230_k = 2 \cdot k^2 + 3 \cdot k^1 + 0 \cdot k^0 = 2k^2 + 3k

Converting all terms to base 10 allows standard algebraic operations to be performed.

Set up the algebraic equation in base 10 and combine like terms.

(k^2 + 3k + 2) + (4k + 3) = 2k^2 + 3k

k^2 + 7k + 5 = 2k^2 + 3k

Equate the sum of the expanded left-hand side terms to the expanded right-hand side.

Rearrange the terms to form a standard quadratic equation

ak^2 + bk + c = 0

2k^2 + 3k - (k^2 + 7k + 5) = 0

k^2 - 4k - 5 = 0

Subtract

k^2 + 7k + 5

from both sides to set the quadratic expression equal to zero.

Factor the quadratic equation to find the possible values of

k

(k - 5)(k + 1) = 0 \implies k = 5 \text{ or } k = -1

Solve for

k

using factorization.

Select the valid base based on the domain of number bases.

k = 5

(since a number base must be a positive integer greater than the largest digit 4 present in the expression).

Base numbers cannot be negative or smaller than any of their constituent digits.

Number Bases and Conversions

Estimated Time:1m 30s

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