MathCounts Training – Number Theory – 5

________ Let $(a\times b\times c)\div(a+b+c)=341$ to be an equation where $a$, $b$ and $c$ are consecutive positive integers. What is the least possible value of $a$? $$ $$
________ The letters $A$, $B$, $C$, $D$, $E$ and $F$ represent digits and $ABC,DEF$ represents a positive six-dight integer. What is the number $ABC,DEF$ if $$4(ABC,DEF)=3(DEF,ABC)$$ $$ $$
________ Jan is thinking of a positive integer. Her integer has exactly 16 positive divisors, two of which are 12 and 15. What is Jan’s number? $$ $$
________ Mady has an infinite number of balls and empty boxes available to her. The empty boxes, each capable of holding four balls, are arranged in a row from left to right. At the first step, she places a ball in the first box (the leftmost box) of the row. At each subsequent step, she places a ball in the first box of the row that still has room for a ball and empties any boxes to its left. How many balls in total are in the boxes as a result of Mady’s $2010^{th}$ step? $$ $$
________ The product of a set of positive integers is 144. What is the least possible sum of this set of positive integers? $$ $$