<s>[INST] Find the least significant digit Explain why the number $(3^{27}\cdot7^{313}\cdot11^{121})^{1000}$ has 1 as its least significant digit. I know I am supposed to use $\pmod{10}$ but I am not sure how to combine the answers of the individual powers and then raise them to $1000$
 [/INST] Notice that $3^3=27\equiv 7\pmod{10}$, $3^4\equiv 21\equiv 1\pmod{10}$. $7^2\equiv 9\pmod{10}$, $7^4\equiv 9^2\equiv 1\pmod{10}$. Also, $11^4\equiv 1^4\equiv 1\pmod{10}$. Now powers distribute over products, so
$$(3^{27}7^{313}11^{121})^{1000}=3^{27000}7^{313000}11^{121000}$$
and since the exponents are all multiples of $4$ we have that each term in the product is congruent to $1$ mod $10$, hence so is the product of all of them.
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