Find the last digit of (K)

Dec 2015
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Earth
Find the last digit of natural number \(\displaystyle (K)\) if
\(\displaystyle K=n 11^n\) ; \(\displaystyle n \in N\)
 

v8archie

Math Team
Dec 2013
7,713
2,682
Colombia
It's the last digit of $n$, quite trivially.
 
Dec 2015
1,085
169
Earth
Let \(\displaystyle \ell{\{T\}}\) be the last digit of \(\displaystyle T\)
\(\displaystyle \ell{\{K\}}=\ell{\{n11^{n}\}}=\ell{\{n\}}\ell{\{11^n\}}\)
\(\displaystyle \ell{\{K\}}=\ell{\{n\}}=n-10int(\frac{n}{10})=10frac{\{\frac{n}{10}\}}\) ;; frac means fractional part
 

v8archie

Math Team
Dec 2013
7,713
2,682
Colombia
It's significantly cleaner to work modulo 10.