UNO High School Problem of the Week Competition
Problem 13
Let the product (12)(15)(16), each factor written in base b, equal 3146 in base b. Let s = 12 + 15 + 16, each term expressed in base b. Then s, in base b is?
Solution
Let P = (b + 2)(b + 5)(b + 6) = 3146(base b)
Therefore b^3 + 13b^2 + 52b + 60 = 3b^3 + b^2 + 4b + 6.
0 = b^3 - 6b^2 - 24b - 27 and b = 9.
But s = (b + 2) + (b + 5) + (b + 6) = 3b + 9 + 4.
There s = 3 * 9 + 9 + 4 = 4 * 9 + 4 = 44(base b).
