When numbers are written in base b, we have (12)b × (25)b = (333)b. What is…

Concept: In base b positional notation, each digit of a number is multiplied by the place-value power of b at its position and the results are added — a two-digit base-b number "XY" equals X×b + Y, and a three-digit base-b number "XYZ" equals X×b2 + Y×b + Z. Also, a base must be strictly greater than every digit that appears in the numbers — since the digit 5 is used here, b must be at least 6.

Application:

1. Expand every base-b number using the place-value rule above: (12)b = b + 2, (25)b = 2b + 5, and (333)b = 3b2 + 3b + 3.

2. Set up the equation from the given product: (b + 2)(2b + 5) = 3b2 + 3b + 3.

3. Expand the left-hand side: 2b2 + 5b + 4b + 10 = 2b2 + 9b + 10.

4. Equate both sides and simplify: 2b2 + 9b + 10 = 3b2 + 3b + 3, which rearranges to b2 − 6b − 7 = 0.

5. Factor the quadratic: (b − 7)(b + 1) = 0, so b = 7 or b = −1.

6. Reject b = −1 (a base cannot be negative); since b must be at least 6, b = 7 is the valid solution.

Cross-check: substituting b = 7 back into the original numbers confirms it — (12)7 = 9 and (25)7 = 19, so 9 × 19 = 171; and (333)7 = 3(49) + 3(7) + 3 = 171. Both sides match, confirming b = 7.