What is the minimum number of ..?
Usually questions which states "minimum number of" or phrases like "atleast" they usually in most cases give us a hint that we should try out pigeonhole principle.
Okay, so we want minimum ordered pair of integers such that
for any pair (a,b) and (c,d)
a≡cmod3 and b≡dmod5
holds.
mod 3 has 0,1,2 as residues or congruence classes.
mod 5 as 0,1,2,3,4 as residues or congruence classes,
Now a can be congruent to any of 0 or 1 or 2 (3 choices)
And b can be congruent to any of 0 or 1 or 2 or 3 or 4 (5 choice)
a can be chosen independently of b and vice versa
so total ways of ordered pair such formed = 3*5=15.
Now if, you take any two ordered pairs from these 15 pairs say (a,b) and (c,d)
It will always be the case that a!=c and b!=d or a=c but b!=d or a!=c but b=d in which case our condition
a ≡ c mod3 and b ≡ d mod5 will violate.
Now if you seem to add one more ordered pair to these 15 pairs(16 or more ordered pairs)
then by pigeon hole principle it will be the case that
a=c and b=d for atleast 2 ordered pairs (a,b) and (c,d) will hold.
in which case our condition
a≡cmod3 and b≡dmod5
will get satisfied.
Hence, the answer is 16.