Question 2

A number $$N$$ in base 10, is 503 in base $$b$$ and 305 in base $$b+2$$. What is the product of the digits of $$N$$?

Correct Answer: 64

Solution

The two representations give $$5b^2 + 3 = 3(b+2)^2 + 5$$, which simplifies to $$2b^2 - 12b - 14 = 0$$, that is $$b^2 - 6b - 7 = 0$$. The positive root is $$b = 7$$, so $$N = 5 \times 49 + 3 = 248$$. The product of its digits is $$2 \times 4 \times 8 = 64$$.