The smallest positive integer that leaves remainder 3 on division by 7 and remainder 4 on division by 9 is:

Numbers leaving remainder 4 on division by 9 are 4, 13, 22, 31, 40, … Test each against the first condition: 4, 13 and 22 leave 4, 6 and 1 on division by 7, but $31=7\times 4+3$ works. So 31 is the smallest; the next one is 31 + 63 = 94, since the pattern repeats every LCM(7, 9) = 63.