Find the last digit of 541^(341). Clearly, the last digit would be one, but I don't know how to go about showing that it would be one. Can someone help please?

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- Nov 20th 2009, 11:51 AMsteph3824Find the last digit
Find the last digit of 541^(341). Clearly, the last digit would be one, but I don't know how to go about showing that it would be one. Can someone help please?

- Nov 20th 2009, 12:06 PMgmatt
$\displaystyle 541^{341} \equiv 1^{341} \equiv 1 \bmod{10} $ i.e. the last digit is going to be 1. If this doesn't make sense to you prove it by induction. I.e. prove that $\displaystyle 541^n $ has ending digit one for all n.

- Nov 20th 2009, 01:24 PMmosta86
by induction : 541^n ,

for n = 1 , 541^1 = 541 => it ends with 1 true

assume that 541^k ends with one is true and proof that 541^(k+1) ends with one

now 541^(k+1) = 541^k * 541 ,,, we prooved that 541^k ends with one and 541 ends with one the multiplication of two numbers that ends with one will result in a number that ends with one .