## 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 \(M(8)\)? | |
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Part B | 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 \(S\)? | |
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Part B | If \(n=5\) then what is the minimum possible value of \(S\)? |