Linear combination
A linear combination of two integers is the sum of their multiples.
The linear combination of two integers \(x\) and \(y\), where \(a\) and \(b\) are any integer
coefficients is:
$$ax + by$$
⚠ If an integer divides two integers, then it can divide any of
their linear combinations. In particular, if \(z \mid x\) and \(z \mid y\), then \(z \mid ax + by.\) Not
convinced? Go prove it with the definition of divides!
To give a specific example, \(5*2 + 7*4\) is a linear combination of \(2\) and \(4.\) It also happens to be a linear combination of \(5\) and \(4\), \(2\) and \(7\), and \(5\) and \(7.\)
Logic & Proofs
Integer •
Rational number •
Inequality •
Real number •
Theorem •
Proof •
Statement •
Proof by exhaustion •
Universal generalization •
Counterexample •
Existence proof •
Existential instantiation •
Axiom •
Logic •
Truth •
Proposition •
Compound proposition •
Logical operation •
Logical equivalence •
Tautology •
Contradiction •
Logic law •
Predicate •
Domain •
Quantifier •
Argument •
Rule of inference •
Logical proof •
Direct proof •
Proof by contrapositive •
Irrational number •
Proof by contradiction •
Proof by cases •
Summation •
Disjunctive normal form
Set Theory
Set •
Element •
Empty set •
Universal set •
Subset •
Power set •
Cartesian product •
String •
Binary string •
Empty string •
Set operation •
Set identity •
Set proof
Functions
Algorithms
Relations
Number Theory
Induction
Combinatorics
Graph Theory
Graph •
Walk •
Subgraph •
Regular graph •
Complete graph •
Empty graph •
Cycle graph •
Hypercube graph •
Bipartite graph •
Component •
Eulerian circuit •
Eulerian trail •
Hamiltonian cycle •
Hamiltonian path •
Tree •
Huffman tree •
Substring •
Forest •
Path graph •
Star •
Spanning tree •
Weighted graph •
Minimum spanning tree •
Greedy algorithm •
Prim's algorithm
Recursion