Question

determine solution of the diophantine equation 54x+21y=906​

Answer 1

$\large\underline{\sf{Solution-}}$

The given equation is

$\rm :\longmapsto\:54x + 21y = 906 - - - (1)$

Now,

➢ Prime Factorization of 54 = 3 × 3 × 3 × 2

➢ Prime Factorization of 21 = 3 × 7

So, it implies,

➢ HCF ( 54, 21 ) = 3

And now, 906 is divisible by 3 and we get 302.

$\bf :\implies\:54x + 21y = 906 \: is \: diophantine \: equation.$

Now,

$\rm :\longmapsto\:54 = 21 \times 2 + 12$

$\rm :\longmapsto\:21 = 12 \times 1 + 9$

$\rm :\longmapsto\:12 = 9 \times 1 + 3$

$\rm :\longmapsto\:9 = 3 \times 3 + 0$

Now,

➢Conversely,

$\rm :\longmapsto\:3 = 12 - 9$

$\rm :\longmapsto\:3 = 12 - (21 - 12)$

$\rm :\longmapsto\:3 = 12 \times 2 - 21$

$\rm :\longmapsto\:3 = (54 - 21 \times 2) \times 2 - 21$

$\rm :\longmapsto\:3 = 54 \times 2 - 21 \times 4 - 21$

$\rm :\longmapsto\:3 = 54 \times 2 - 21 \times 5$

Multiply by 302 on both sides, we get

$\rm :\longmapsto\:906 = 54 \times 604 - 21 \times 1510$

So,

$\rm :\implies\:x_0 = 604 \: \: \: and \: \: \: y_0 = - \: 1510$

Hence, general solution is

$\rm :\longmapsto\:x = 604 + 7k$

and

$\rm :\longmapsto\:y = - 1510 - 18k$

➢ Where, k is integer.

Additional Information :-

Diophantine Equation :-

The Diophantine equation was first study by Diophantus of Alexandria, a 3rd century mathematician who also introduced symbolisms into algebra. He was author of a series of books called Arithmetica.

The Diophantine equation is of the form ax + by = c, where a, b and c are given integers.

This Diophantine equation has a solution, where x and y are integers iff c is a multiple of the highest common factor of a and b.