Find the least positive integer x that satisfies 90x 41 mod

Find the least positive integer x that satisfies 90x 41 (mod 73) ?????

I have found that the gcd(90,73) is 1 and by running Euclid\'s Alg. found that 1 = 90(-30) + 73(37) so 90(-1230) + 73(1517) = 41. So any integer congruent to -1230(mod 73) is a solution, but since I need the least POSITIVE integer, I tried -1230 = 73(-16) + (-62) but -62 is obviously still negative so I wasn\'t sure what to do from here, but this is what I tried: 73 = -62(-1) + (11) thus 11 would be the smallest positive integer so x=11. But I don\'t know if that\'s actually mathematically aloud/correct. Please help!

Solution

yes 11 is correct answer