Question 25

Let the product $$12 \times 15 \times 16 = 3146$$ in base $$b$$, where $$b$$ is a real number. Then the sum $$s = 12 + 15 + 16$$ in base $$b$$ is

Correct Answer: 44

Solution

In base $$b$$ the factors are $$b + 2$$, $$b + 5$$, $$b + 6$$ and the product is $$3b^3 + b^2 + 4b + 6$$, so $$b^3 + 13b^2 + 52b + 60 = 3b^3 + b^2 + 4b + 6$$. This reduces to $$b^3 - 6b^2 - 24b - 27 = 0$$, which is satisfied by $$b = 9$$. The sum is $$3b + 13 = 40$$ in decimal, and $$40 = 4 \times 9 + 4$$, so in base 9 the sum is written as 44.

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