If 203_n - 114_n = 45_n, where n represents a positive integer base, what is the value of … — JAMB UTME

If $203_n - 114_n = 45_n$ , where $n$ represents a positive integer base, what is the value of $n$ ?

The correct base is

6

Expanding the numbers in terms of powers of

n

gives

(2n^2 + 3) - (n^2 + n + 4) = 4n + 5

. Grouping like terms results in the quadratic equation

n^2 - 5n - 6 = 0

. Factoring yields

(n - 6)(n + 1) = 0

, giving

n = 6

as the only valid positive integer solution greater than

5

Convert each term from base

n

to base 10 using positional expansion.

203_n = 2n^2 + 0n + 3 = 2n^2 + 3

114_n = 1n^2 + 1n + 4 = n^2 + n + 4

, and

45_n = 4n + 5

Converting all terms to a common decimal representation allows algebraic manipulation.

Substitute the expanded terms back into the original equation and simplify.

(2n^2 + 3) - (n^2 + n + 4) = 4n + 5 \implies n^2 - n - 1 = 4n + 5

Carefully distribute the negative sign across all terms of

(n^2 + n + 4)

Rearrange the expression into standard quadratic form

ax^2 + bx + c = 0

n^2 - 5n - 6 = 0

Subtracting

4n + 5

from both sides sets the quadratic equation to zero.

Factor the quadratic equation to find the valid base

n

(n - 6)(n + 1) = 0 \implies n = 6

n = -1

Since a base must be a positive integer strictly greater than any digit present in the equation (maximum digit is

5

n = 6

Solving unknown base equations using polynomial expansion in positional notation.

Question