<s>[INST] Last digit on $3^{100}$ How to find the last digit on $3^{100}$?
Is there any proper method to solve such questions without calculator of course?
 [/INST] Solving questions like these are pretty simple once you know a certain trick:
Let's work out the first few terms of $x = 3^n$


*

*When n = 0: | x = 1 | The last digit is 1

*When n = 1: | x = 3 | The last digit is 3

*When n = 2: | x = 9 | The last digit is 9

*When n = 3: | x = 27 | The last digit is 7

*When n = 4: | x = 81 | The last digit is 1

*When n = 5: | x = 243 | The last digit is 3
There is a pattern! 1,3,9,7,1,3,9,7,1,3,9,7,1,3,9,7,1,3,9,7... and so on. The pattern repeats itself every 4 iterations. {1,3,9,7}{1,3,9,7}...
We can therefore deduce that if:
($n$ mod 4) = 0, The last digit is 1
($n$ mod 4) = 1, The last digit is 3
($n$ mod 4) = 2, The last digit is 9
($n$ mod 4) = 3, The last digit is 7

To work out the final answer of this, let's see what happens when n = 100
$100 $ mod $4 = 0$, Therefore the last digit is a 1! Problem solved!
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