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Discrete Mathematics

Perform basic number theory operations such as factoring:

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⨯ 
FactorInteger[30]
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Find the GCD (or LCM) of any two integers:

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⨯ 
GCD[24, 60]
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Display the 4th prime number:

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⨯ 
Prime[4]
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Test the primality of a number:

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⨯ 
PrimeQ[%]
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This works with coprime numbers as well:

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⨯ 
CoprimeQ[51, 15]
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Use the Mod function for the remainder:

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⨯ 
Mod[17, 5]
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Get all possible permutations of a list:

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⨯ 
Permutations[{a, b, c}]
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Apply Permute to a list using disjoint Cycles:

(Cycles takes a list of lists as an argument.)
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⨯ 
Permute[{a, b, c, d}, Cycles[{{2, 4}, {1, 3}}]]
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Find the permutation order:

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⨯ 
PermutationOrder[Cycles[{{2, 4}, {1, 3}}]]
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Generate a Graph from a list of edges:

(Use ESCueESC for an UndirectedEdge or ESCdeESC for a DirectedEdge.)
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⨯ 
Graph[{1 <-> 2, 2 \[DirectedEdge] 3, 3 \[DirectedEdge] 4, 4 <-> 1, 
  3 \[DirectedEdge] 1, 2 \[DirectedEdge] 2}, VertexLabels -> All]
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Find the shortest path between two vertices:

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⨯ 
FindShortestPath[%, 3, 2]
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Explore well-known graphs using natural-language input:

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pappus graph image
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The Wolfram Language also includes functions for combinatorics, probability, integer sequences and much more.

QUICK REFERENCE: Number Theoretic Functions »

QUICK REFERENCE: Discrete Mathematics »