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you will notice that the numbers 11, 23 are common between the 2 statements..go ahead and try you will see that every 12 numbers apart you will have a number that satisfies both statement 1 and 2...

it can be checked whether together they are sufficient:

4x+3 = n

when x = 1 2 3 4 5 6 7 8...
than remainder = 1 5 3 1 5 3 1 5 ... (1 5 3 repeats)

3x+5 = n
when y = 1 2 3 4 5 6 7 8 ...
then n = 2 5 2 5 2 5 2 5 ... (2 5 repeats)

there are some numbers as 2, 8, 14, 20 ... in which remainder is the same for both equations (these numbers are just plug ins) only these so called 'plug ins' satisfy both equations at the same time.