When positive integer x is divided by 5, the remainder is 2.

One way to solve this problem:-
since x=5a+2, where a is an integer.
and y=4b+1 where b is an integer
x+y=5a+4b+3..

now test for integer solutions of a and b. We need to subtract 3 from answer choice and then test whether the resultant can be expressed 5a+4b.
choice a) 12. Subtract 3 we get 9 . 9 = 4a+5b. take a=1, b=1.
choice b) 13. Subtract 3 we get 10. 10=4a+5b. take a=0 and b=2.
choice c) 14. Subtract 3 we get 11. 11 = 4a+5b. if we take a=0, 1,2, can we have b which is positive integer. let us check
a=0 ;11=5b , a = 1 ; 7 = 5b, a = 2 ; 3= 5b, a = 3 , now in this case -1 = 5b.. now from here on we can get negative values on the L.H.S. So we cannot express 11 = 4a+5b where a and b are integers. This is our answer.
choice d) 16 . Subtract 3 we get 13 . 13= 4a+5b. take a =2 and b = 1.
choice e) 21. subtract 3 we get 18. 18= 4a+5b. take a=2, b=2. Done!