- #1

Math100

- 792

- 220

- Homework Statement
- Using congruences, solve the Diophantine equation below:

## 4x+51y=9 ##.

[Hint: ## 4x\equiv 9\pmod {51} ## gives ## x=15+51t ##, whereas ## 51y\equiv 9\pmod {4} ## gives ## y=3+4s ##. Find the relation between ## s ## and ## t ##.]

- Relevant Equations
- None.

Consider the Diophantine equation ## 4x+51y=9 ##.

Observe that ## gcd(51, 4)=1 ##.

Then ## 4x\equiv 9\pmod {51}\implies 52x\equiv 117\pmod {51}\implies x\equiv 15\pmod {51} ##.

Now we have ## 51y\equiv 9\pmod {4}\implies 3y\equiv 1\pmod {4}\implies y\equiv 3\pmod {4} ##.

This means ## x=15+51t ## and ## y=3+4s, \forall t, s ##.

Since ## 4(15+51t)+51(3+4s)=9 ##, it follows that ## s=-1-t ##.

Thus ## y=3+4s=3+4(-1-t)=-1-4t ##.

Therefore, ## x=15+51t ## and ## y=-1-4t, \forall t, s ##.

Observe that ## gcd(51, 4)=1 ##.

Then ## 4x\equiv 9\pmod {51}\implies 52x\equiv 117\pmod {51}\implies x\equiv 15\pmod {51} ##.

Now we have ## 51y\equiv 9\pmod {4}\implies 3y\equiv 1\pmod {4}\implies y\equiv 3\pmod {4} ##.

This means ## x=15+51t ## and ## y=3+4s, \forall t, s ##.

Since ## 4(15+51t)+51(3+4s)=9 ##, it follows that ## s=-1-t ##.

Thus ## y=3+4s=3+4(-1-t)=-1-4t ##.

Therefore, ## x=15+51t ## and ## y=-1-4t, \forall t, s ##.