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Number Theory

# Last Digits

What is the last digit of $123 \times 321 ?$

Hint: There is no need to completely multiply these numbers.

All of the problems in this quiz can be solved without completing full calculations if you focus in on just the part that matters. This kind of shortcut-seeking is a major skill you'll practice and explore as you learn number theory.

Some of the number theory tricks shortcuts in this course are purely theoretical, and others can be applied to doing anything from rapidly computing what day of the week your $$100^\text{th}$$ birthday will fall on to hacking smartphone software!

What is the last digit of

$1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 ?$

What is the last digit of

$1 \times 3 \times 5 \times 7 \times 9 \times 11 \times 13 \, ?$

What is the last digit of $$2^{13} ?$$


Hint: \begin{alignat*}{2} &2^1 = \, &\color{red}2\\ &2^2 = \, &\color{red}4\\ &2^3 = \, &\color{red}8\\ &2^4 = \, &1\color{red}6\\ &2^5 = \, &3\color{red}2\\ &2^6 = \, &6\color{red}4\\ &2^7 = \, &12\color{red}8\\ &2^8 = \, &25\color{red}6 \end{alignat*}

What is the last digit of $$6^{789} ?$$


Hint:

• $$6 \times 6 = 3\color{red}6$$
• $$36 \times 6 = 21\color{red}6$$
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