If 24_x \times 13_x = 345_x, where x represents a positive integer base, find the value of… — JAMB UTME

If $24_x \times 13_x = 345_x$ , where $x$ represents a positive integer base, find the value of $x$ .

Answer: 7

The value of the base

x

is 7.

Expanding

24_x

13_x

, and

345_x

into base 10 yields

(2x + 4)(x + 3) = 3x^2 + 4x + 5

. Expanding the left side gives

2x^2 + 10x + 12

. Equating and simplifying gives

x^2 - 6x - 7 = 0

, which factors as

(x - 7)(x + 1) = 0

. The positive integer solution greater than 5 is

x = 7

Convert all base

x

numbers to decimal (base 10) expressions.

24_x = 2x + 4

13_x = x + 3

, and

345_x = 3x^2 + 4x + 5

Place-value expansion allows algebraic manipulation in standard base 10.

Multiply the expanded factors on the left-hand side.

(2x + 4)(x + 3) = 2x^2 + 10x + 12

Applying the distributive property of multiplication.

Equate the expanded left-hand side to the right-hand side and rearrange into standard quadratic form.

3x^2 + 4x + 5 - (2x^2 + 10x + 12) = 0 \implies x^2 - 6x - 7 = 0

Setting the quadratic expression equal to zero enables factoring.

Factor the quadratic equation and select the valid base.

(x - 7)(x + 1) = 0 \implies x = 7

(rejecting

x = -1

A number base must be a positive integer strictly greater than any individual digit in the given numbers (max digit is 5).

Solving polynomial equations derived from number base expansion.

Question