MathCounts Training – Number Theory – 4

________ What is the greatest positive integer that must divide the sum of the first ten terms of any arithmetic sequence whose terms are positive integers?$$ $$
________ A digit can be placed in each of the boxes for hundreds and units digits to form the least positive five-digit number divisible by 96. What is the ratio between the smaller digit to the larger digit? Express your answer as a common fraction. $$21\boxed{\phantom{8} }7\boxed{\phantom{8}}$$
________ In order, the first four terms of a sequence are 2, 6, 12 and 72, where each team, beginning with the third term, is the product of the two proceeding terms. If the ninth term is $2^a3^b$, what is the value of $a+b$? $$ $$
________ What is the sum of the numbers less than 200 that have exactly 9 divisors? $$ $$
________ Given that $A=\{\dfrac{n}{24}\}$, $n$ is a natural number, $gcd(n,24)=1$, and $\dfrac{n}{24}\lt 2$, what is the sum of the elements in $A$? $$ $$