Consider the equation \((123)_5 = (x8)_y\) with \(x\) and \(y\) as unknown.…

Convert the base-5 number to decimal:

(123) in base 5 equals 1×25 + 2×5 + 3 = 25 + 10 + 3 = 38

We need a base-y two-digit number with digits x and 8 to equal 38, so:

x·y + 8 = 38, therefore x·y = 30.

Constraints:

• The base y must be greater than 8 (so that digit 8 is valid).

• x must be an integer with 0 ≤ x < y.

Find integer factor pairs of 30 with y > 8 and x < y:

• x = 1, y = 30 → valid (30 > 8 and 1 < 30).

• x = 2, y = 15 → valid (15 > 8 and 2 < 15).

• x = 3, y = 10 → valid (10 > 8 and 3 < 10).

Other factor pairs either have base y ≤ 8 or do not satisfy x < y, so they are invalid.

Therefore, the number of possible solutions is 3.