Don't know how to continue modulo reduction

Question

Ok , so the question is to find the last three digits of $2013^{2012}$. After some reduction using Euler's Theorem I got $13^{12}$(mod 1000). I tried dividing it in 8 and 125 and later use the CRT , but this didn't help (or maybe I did it wrong) . What is the shortest way to get the answer ? (ofc. without using very big numbers)

Answer 1

$$13^{12}=(13^2)^6=(170-1)^6\equiv1-\binom61170+\binom62(170)^2\pmod{1000}$$

$$\implies13^{12}\equiv1-6\cdot170+\frac{6\cdot5}2\cdot289\cdot100\pmod{1000}$$

Now $\displaystyle \frac{6\cdot5}2\cdot 289=15\cdot289\equiv5\pmod{10}$

Using $\displaystyle a\equiv b\pmod m\implies a\cdot c\equiv b\cdot c\pmod{m\cdot c},$ (where $a,b,c,m$ are integers) $\displaystyle\implies\frac{6\cdot5}2\cdot289\cdot100\equiv5\cdot100\pmod{10\cdot100} $

$\displaystyle\implies13^{12}\equiv1-1020+500\equiv501-20\pmod{1000}$

This method can be applied for any number co-prime with $10$(hence with $10^k$)

as we know, any number co-prime with $10$(hence with $10^k$) must end with $1,3,7,9$

and $7^2=49=5\cdot10-1$ and $3^2=9=10-1$

Answer 2

HINT:

$$13^{12}=(10+3)^{12}=3^{12}+12\cdot3^{11}10+\binom{12}2\cdot3^{10}10^2\pmod{1000}$$

Now, $\displaystyle3^{10}=9^5\equiv(-1)^5\pmod{10}\equiv-1$

$\displaystyle\implies\binom{12}2\cdot3^{10}10^2\equiv\binom{12}2\cdot(-1)10^2\pmod{100}$

and $\displaystyle3^{11}=3\cdot9^5=3(10-1)^5\equiv3(-1+\binom51\cdot10^1)\pmod{100}$

Finally, $3^{12}=(3^2)^6=9^6=(10-1)^6=1-\binom6110+\binom6210^2\pmod{1000}$

Answer 3

Even without binomial tricks, this isn't too bad; just remember to produce intermediate results. $13^2=169$ and $13^3$ $=13\cdot 169$ $= 1690+507$ $\equiv 197$ are more or less immediate, and then this just needs to be squared twice. At this point you can either do it 'by hand' or go with $197=200-3$ to get $13^6$ $\equiv 197^2$ $= 200^2-2\cdot3\cdot200+9$ $\equiv 9-200$ $\equiv 809$ and $13^{12}$ $\equiv 809^2$ $= (800+9)^2$ $= 800^2+2\cdot 9\cdot 800 + 9^2$ $\equiv 481$.

Answer 4

${\rm mod}\ \ \ \ \ 8\!:\ 13^{12}\equiv (-3)^{12}\!\equiv 9^6\equiv 1^6\equiv \color{#c00}1\ \ \ \ $ [All arithmetic in this answer can be done mentally] ${\rm mod}\ 125\!:\ 13^{12}\!\equiv 44^6 \equiv 4^6 11^6 \equiv 2^{12} (-4)^3\equiv -2^{18}\overset{\large 2^7\,\equiv\ \color{#c0f}3\ }\equiv\!\! -\color{#c0f}3^2(2^4)\equiv\, \color{#0a0}{-19}$

By Easy CRT: ${\ \ 13^{12} \equiv\, \color{#0a0}{{-}19} + 125\underbrace{\left[\dfrac{\color{#c00}1+\color{#0a0}{19}}{125}\ {\rm mod}\ 8\right]}_{\large \equiv\, 20/5\,\equiv\, 4\qquad}\equiv\, {-}19+125(4) \equiv 481\pmod{1000}}$

Remark $\ $ I'd choose the binomial theorem method if I were evaluating it, but it's worth emphasis that CRT is only slightly more work (and good CRT practice). With a little practice one can tackle such small CRT problems with only mental arithmetic of small numbers (as I did above).