Least-Significant Decimal digit I have an exam coming up an this will be one style of question can anyone please walk me through how it is done?
What is the least-significant decimal digit of $1002^{3755}$?
 A: The idea is: 
You need to calculate mod 10. And since you only need mod 10, you can restrict to $2^{3755}$. 
And that is easy, because the results of the power of 2 end on 2,4,8,6,2,4,8,6, etc. 
So you see, every 4th time you find the same value. Therefore the exponent 3755 has the same last digit as the exponent 3 (since 3755 = 3 mod 4) which is 8. 
The standard way to calculate $2^{3755}$ would be: 


*

*Find that $2^5 = 2 mod 10$. 

*Then $2^{3755} = 2^{5*751} = (2^{5})^{751} = (2)^{751} = (2)^{5*150 +1} = (2^5)^{150} *2=(2)^{5*30} *2=2^{30}*2 = 2^6*2=(2)^{5+1}*2 =(2)^5 *2^1*2= 2*2*2 = 8$


So 8 should be the correct answer.
A: The "least-significant" digit is the rightmost one.  For example, in the numeral 769, the least-significant digit is the 9.  It's "least significant" because it only counts for 9 in determining the value of 769. The 6 is actually worth 60; the 7 is worth 700.
So the question is really asking for the units digit of $1002^{3755}$, which is the same as asking for the residue of $1002^{3755}$ mod 10.  The first thing to do is to observe that $1002^{3755} \equiv  2^{3755}\pmod{10}$.  
Does that help?
