Find the last digit

Dec 2015
1,082
169
Earth
Find the last digit of : \(\displaystyle y=12^{12^{12} } \) .

My first approach is to find the last digit of exponent \(\displaystyle 12^{12}
\;\) , then let n be the last digit of exponent , now the last digit of y is the last digit of \(\displaystyle 12^n\) .
 

romsek

Math Team
Sep 2015
2,959
1,673
USA
$12^{12^{12}} = 12^{144}$

$12^{144} = (10+2)^{144} = \\

\sum \limits_{k=0}^{144}~\dbinom{144}{k}10^k 2^{144-k} =\\

2^{144} + \text{143 terms all divisible by 10}$

$144 = 28 \cdot 5 + 4$

$2^{144} = \left(2^5 \pmod{10}\right)^{28} \cdot (2^4 \pmod{10}) \pmod{10}= \\

((2^{28}\pmod{10})\cdot 6) \pmod{10} = \\6\cdot 6 \pmod{10}=6$
 
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