<s>[INST] How to find the unit digit in the product of exponents? 
What is the unit digit in the product $3547^{153}\times251^{72}$?

I know that the required digit is equal to the unit digit in $7^{153}\times1^{72}$. And I also know that $1^{72}=1$, as $1$ is the multiplicative identity. But I am having trouble evaluating $7^{153}$.
Similarity,

What is the unit digit in $264^{102}+264^{103}$?

$$264^{102}+264^{103}=264^{102}×(1+264)=264^{102}\times265=\cdots$$
 [/INST] $251^{72}  \equiv 1 \pmod{10}$
$3547^{153} \equiv 7^{153} \pmod{10}$
Cyclic sequence of $7^{n}$ is as follows.
$7^{1} = 7$
$7^{2} = 49$
$7^{3} = 343$
$7^{4} = 2401$
Lets divide $153$ by $4$ and the remainder is $1$.
Thus, the unit digit of $7^{153}$ is equal to the unit digit of $7^{1} = 7.$
Hence the unit digit of $3547^{153}\times251^{72}$ is equal to $7.$
$264^{102} \equiv 4^{102} \pmod{10}$
Cyclic sequence of $4^{n}$ is as follows.
$4^{1} = 4$
$4^{2} = 16$
Lets divide $102$ by $2$ and the remainder is $0$.
Thus, the unit digit of $4^{102}$ is equal to the unit digit of $4^{2} = 6.$
Hence the unit digit of $264^{102}+264^{103}$ is equal to $0.$
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