Linear Diophantine Equation/Examples/15x + 51y = 41

Example of Linear Diophantine Equation

The linear diophantine equation:

$15 x + 51 y = 41$

has no solutions for $x$ and $y$ integers.

Graphical Presentation

Proof

Using the Euclidean Algorithm:

\(\text {(1)}: \quad\)

\(\ds 51\)

\(=\)

\(\ds 3 \times 15 + 6\)

\(\text {(2)}: \quad\)

\(\ds 15\)

\(=\)

\(\ds 2 \times 6 + 3\)

\(\text {(3)}: \quad\)

\(\ds 6\)

\(=\)

\(\ds 2 \times 3\)

Thus we have that:

$\gcd \set {15, 45} = 3$

which is not a divisor of $41$.

So, from Solution of Linear Diophantine Equation, no solution exists.

$\blacksquare$

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {2-3}$ The Linear Diophantine Equation: Exercise $1 \ \text {(c)}$