MathCounts Training – Number Theory – 2

________ In base $b$, $441_b$ is equal to $n^2$ in base 10, and $351_b$ is equal to $(n-2)^2$. What is the value of $b$, expressed in base 10?
________ The base-three representation of $0.\overline{12}$ is equivalent to what base-ten common fraction?
________ For how many natural numbers less than 100 is the product of the number’s distinct prime factors equals to 6?
________ A positive number is called $n$-primable if it is divisible by $n$ and each of its digits is a one-digit prime number. How many 3-primable positive integers are then that are less than 1000?
________ Simplify: $$(1\dfrac{1}{2})^{-2}+(1\dfrac{1}{2})^{-1}+(1\dfrac{1}{2})^{0}+(1\dfrac{1}{2})^{1}+(1\dfrac{1}{2})^{2}$$