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I am trying to manage a matrix in Mathematica that has the general architecture

{{a1,b1},{a2,b2}...}

I would like to combine all of the lists that have the same the first value and transform it to a new list of lists of the second values,

For example, If I have the list {{1,2},{1,5},{1,30},{2,6},{3,8},{3,30},{4,1}}

I would like the output to be {{2,5,30},{6},{8,30},{1}}

I would then like to remove any of the list elements that contain only 1 value

So that my final output would be {{2,5,30},{8,30}}

I am still new to data manipulation in mathematica and really do not know where to begin on this problem

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Use GroupBy:

Values @ DeleteCases[{_}] @ GroupBy[
    {{1,2},{1,5},{1,30},{2,6},{3,8},{3,30},{4,1}},
    First->Last
]

{{2, 5, 30}, {8, 30}}

or GatherBy

DeleteCases[{_}] @ GatherBy[
    {{1,2},{1,5},{1,30},{2,6},{3,8},{3,30},{4,1}},
    First
][[All, All, 2]]

{{2, 5, 30}, {8, 30}}

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