I hope u got the question meaning first. Now the task is, we have to find the minimum number of ordered pairs of non-negative numbers which means ordered pairs start from (0,0) to till the last nonnegative number, suppose it is (N, N).
They are giving two expressions which say a ≡ c mod 3 which means a and c will leave the same remainder when we divide them with 3.
Ex: 24 ≡ 12 mod 3, 15 ≡ 18 mod 3
Similarly, another expression has also a similar meaning.
whenever we divide any number with 3, there are 3 possibilities of remainders:(0,1,2) and when we divide the number with 5, there are possibilities of remainders:(0,1,2,3,4).
So we have total 3 *5 = 15 pairs which starts from (0,0),(0,1) .........(2,4)
Now it is becoming like a function in the sense we will give any pair of input( from (0,0) to (N, N)) then we will check it maps to which possibilities among((0,0),(0,1)..............(2,4)).
we can also take this way, there are 15 boxes ; (0,0) - Box 1, (0,1) - Box 2 ,.......(2,4) -Box 15
Now when we take any non-negative ordered pair, we put (0,0) to Box 1, (0,1) to Box 2 ,.....(2,4) to Box 15,
next, when we take any ordered pair now if it is satisfying conditions of expressions, it will sit in one of the 15 Boxes, we have found out, we have to take a minimum of 16 ordered pairs, we can treat these ordered pairs as Pigeons, there will at least 16 Pigeons (ordered pairs).