Given that 123_x = 313_5, what is the value of the base x? — JAMB UTME

Given that $123_x = 313_5$ , what is the value of the base $x$ ?

Converting

313_5

to base 10 yields

3(25) + 1(5) + 3(1) = 83

. Expanding

123_x

yields

x^2 + 2x + 3

. Setting them equal produces

x^2 + 2x + 3 = 83

, which simplifies to

x^2 + 2x - 80 = 0

. Solving

(x-8)(x+10) = 0

gives

x = 8

since a base must be positive.

Convert the right-hand side

313_5

to base 10.

3(5^2) + 1(5^1) + 3(5^0) = 75 + 5 + 3 = 83_{10}

Converting all terms to base 10 allows forming a standard algebraic equation.

Expand the left-hand side

123_x

in powers of

x

1 \cdot x^2 + 2 \cdot x^1 + 3 \cdot x^0 = x^2 + 2x + 3

Expressing the number in terms of its base

x

positional values.

Set the two base 10 expressions equal and solve the quadratic equation.

x^2 + 2x + 3 = 83 \implies x^2 + 2x - 80 = 0 \implies (x + 10)(x - 8) = 0 \implies x = 8

The base

x

must be a positive integer greater than any digit in

123_x

(so

x > 3

), which leaves

x = 8

Solving equations with unknown number bases by expanding into base 10 polynomials.

Soru