- #1

knowLittle

- 312

- 3

## Homework Statement

Find all solutions to the equation ##35x\equiv 10mod50##

## The Attempt at a Solution

gcd( 35,50)= 5

So, there is a solution to this, since 5| 10. Also, there is a theorem that guarantees the existence of exactly 5 solutions.

Now, dividing ##35x\equiv 10mod50## over 5 gives:

##7x\equiv 2mod10##

Now, what multiple of 7 gives us ##\equiv 2mod10##

{ 2, 12, 22, 32, 42,...} Here, we found 42 that is a multiple of 7 and satisfies ##\equiv 2mod10##

We can write ##7x\equiv 42mod10## . Now, I divided the expression by 7 and got ##x\equiv 6mod10##

Now, there is another theorem that tells me this

6+(50/5)t, t=0, 1, ..., 4

I get: 6, 16, 26, 36, 46

So, the solutions are

##x\equiv 6mod50## , ##x\equiv 16 mod50##, ##x\equiv 26 mod50##, ##x\equiv 36mod 50##, ##x\equiv 46mod 50##

These are the 5 solutions of ##35x\equiv 10mod50##

I found other solutions online:

So x = 6, 16, 22, 28, 34, 40, 46 modulo 50 are the

solutions to the congruence 35x ≡ 10 mod 50.

Am I incorrect?

Thank you.