Question 632981: When you square any integer, the last digit of the number you get will never be an eight.
Is this statement true, false, or sometimes true. Justify your result.
to prove this i've tried...
1^2=1 2^2 = 4, 3^2 = 9, 4^2 = 16, 5^2 = 25, 6^2=36, 7^2=49, 8^2=64, 9^2=81, 10^2 = 100
as you can see this is true for single digits i've tried double digits as well up till 30^2, and clearly enough the last digit is never an eight.
from what i've got above im guessing that this statement is true!
Found 2 solutions by Edwin McCravy, richard1234:
Answer by Edwin McCravy(20054)   (Show Source):
You can put this solution on YOUR website!
Every integer ends with either a 0,1,2,3,4,5,6,7,8, or 9.
The last digit of the square of a number that ends with 0, ends with 0×0 or 0
The last digit of the square of a number that ends with 1, ends with 1×1 or 1
The last digit of the square of a number that ends with 2, ends with 2×2 or 4
The last digit of the square of a number that ends with 3, ends with 3×3 or 9
The last digit of the square of a number that ends with 4, ends with what 4×4 or 16 ends with, namely 6.
The last digit of the square of a number that ends with 5, ends with what 5×5 or 25 ends with, namely 5.
The last digit of the square of a number that ends with 6, ends with what 6×6 or 36 ends with, namely 6.
The last digit of the square of a number that ends with 7, ends with what 7×7 or 49 ends with, namely 9.
The last digit of the square of a number that ends with 8, ends with what 8×8 or 64 ends with, namely 4.
The last digit of the square of a number that ends with 9, ends with what 9×9 or 81 ends with, namely 1.
So squares can only end with 0,1,4,5,6, and 9
Therefore squares cannot end with 2,3,7, or 8.
Edwin
Answer by richard1234(7193)  (Show Source):
You can put this solution on YOUR website! You only need to check up to 10^2, because the last digit will repeat (because 12 is congruent to 2 mod 10, for example). I would brute force just like the other tutor did to check that a perfect square cannot end in 8.
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