# Euclidean Algorithm/Examples/341 and 527/Integer Combination

## Examples of Use of Euclidean Algorithm

$31$ can be expressed as an integer combination of $341$ and $527$:

$31 = 2 \times 527 - 3 \times 341$

Note also that:

$31 = 14 \times 341 = 9 \times 527$

and:

$31 = 13 \times 527 - 20 \times 341$

## Proof

From Euclidean Algorithm: $341$ and $527$ we have:

$\gcd \set {341, 527} = 31$

Then we have:

 $\ds 31$ $=$ $\ds 186 - 1 \times 155$ from $(3)$ $\ds$ $=$ $\ds 186 - 1 \times \paren {341 - 1 \times 186}$ from $(2)$ $\ds$ $=$ $\ds 2 \times 186 - 1 \times 341$ $\ds$ $=$ $\ds 2 \times \paren {527 - 1 \times 341} - 1 \times 341$ from $(1)$ $\ds$ $=$ $\ds 2 \times 527 - 3 \times 341$

$\blacksquare$