Ateacher asked 3 pupils to choose any 2 digit number square it and then find the remainder after dividing the result by 7. doris got 6 marie got 4, horace got 3. two of the pupils made a mistake. why?

First write out all the two digit square numbers

4² = 16
5² = 25
6² = 36
7² = 49
8² = 64
9² = 81

Next divide all of them by 7

16/7 = 2 2/7 or 2 and 2 remainders
25/7 = 3 4/7 or 3 and 4 remainders
36/7 = 5 1/7 or 5 and 1 remainder 
49/7 = 7
64/7 = 9 1/7 or 9 and 1 remainder
81/7 = 11 4/7 or 11 and 4 remainders

This tells us that the only possible remainders for a two digit square number divided by 7 are 1, 2 and 4
If Doris got 6 and Horace got 3, they must have made a mistake.