MathCounts Training – Number Theory – 1

________ Some positive integers have exactly four positive factors. For example, 35 has only 1, 5, 7, and 35 as its factors. What is the sum of the smallest five positive integers that have exactly four factors.
________ What is the largest integer value of $n$ for which $8^n$ evenly divides $100!$?
________ What is the greatest prime factor of $12!+14!$? (Reminder: if $n$ is a positive integer, then $n!$ stands for product $1\cdot 2\cdot 3\cdot ……(n-1)\cdot n$.)
________ The base-10 number 217 and 45 are multiplied. The product is then written in base-6. What is the units digit of the base-6 representation.
________ The smallest case of 5 consecutive odd integers whose sum is a perfect square is 1, 3, 5, 7, 9. (1+3+5+7+9=25.) Find the median of the next larger set of 5 consecutive odd integers whose sum is a perfect square.