modulo help

(a) Let's note a pattern in the equation. 

$x=1; 346 \equiv 346\pmod{347}\\ x=2; 346*2 \equiv 345\pmod{347}\\ x=3; 346*3 \equiv 344 \pmod{347}\\ x=4; 346*4 \equiv 343 \pmod{347}\\ etc.$

Thus, we have 

$347-x=129$ as the pattern for this congruence. 

thus, solving for x, we find that $x=218$

Testing if this is indeed true, we find that 

$x=218; 346*218 \equiv 129 \pmod{347}$

So 218 is the answer. 

Thanks! :)

(b) Let's use a different tactic here. 

We want the fact that 4x subtracted by the remainder is divisble by 75. 

We can have the equation $75y=4x-71$, where y is another integer. 

Isolating x from this equation, we get

$x=\frac{75y+71}{4}$

Now, any value of y we plug in should work. We just need an integer, and we're done. 

When we plug in $y=3$, we find that

$x=\frac{75(3)+71}{4}=\frac{296}{4}=74$

Thus, 74 is our answer. Just fit it into the interval! Lol

Thanks! :)