# Computation Workshop Solution Checker

## Distinct Sums

Let $$S$$ be a set of $$n$$ different positive integers such that every pair of integers in $$S$$ have a different sum. We are looking for the minimum value of the sum of all elements in $$S$$. For a fixed integer $$n$$, let $$M(n)$$ denote the minimum possible sum of the elements of $$S$$.

For example $$M(4)=11$$, because the minimal set $$S$$ with $$4$$ elements is $$S = \{ 1,2,3,5 \}$$ and $$1+2+3+5=11$$.

Part A What is $$M(8)$$? What is $$M(12)$$?

## GCD Acrobatics

Let $$n$$ be a given positive integer. Consider a set $$X$$ of $$n$$ different positive integers such that the greatest common divisor of any $$4$$ different members of $$X$$ is $$1$$ however the greatest common divisor of any $$3$$ members of $$X$$ is more than $$1$$. Let $$S$$ be the sum of all the members of $$X$$. For example if $$n=3$$ then the minimum possible value of $$S$$ would be $$12$$ (which can be achieved by the set $$X = \{ 2,4,6 \}$$).

Part A If $$n=4$$ then what is the minimum possible value of $$S$$? If $$n=5$$ then what is the minimum possible value of $$S$$?