I looked at it this way:
N = 23m + 22 (1) m is an integer
N = 21n + 14 (2) n is an integer
From (1) and (2) we have:
21n - 23m = 8
or 21(n-m) -2m = 8
or 21(n-m)/2 - m = 4 (3)
Now (n-m)/2 must be an integer to satisfy (3).
Try (n - m)/2 = 1 (4)
Put this into (3) to get 21*1 - m = 4 or m = 17 (5)
From (4) and (5) n = 2 + m = 2 + 17 = 19 (6)
Put (5) into (1) (or (6) into (2)): N = 23*17 + 22 = 413.
What if (n - m)/2 = 2
21*2 - m = 4 so m = 38. This time n = 4 + m = 42. Hence N = 23*38 + 22 = 896 which is greater than 500.
Clearly, larger values of (n - m)/2 will produce even larger values of N, so N = 413.