1. ## lest significant digit

Hi I was wondering if somebody could tell me how to find the least significant decimal digit of a large number - like how would i step through it and is there and algorithm numbers like this in general
eg 1002^3755

2. Originally Posted by pablo26
Hi I was wondering if somebody could tell me how to find the least significant decimal digit of a large number - like how would i step through it and is there and algorithm numbers like this in general
eg 1002^3755
The least significant digit of $\displaystyle N$ is $\displaystyle N \pmod {10}$.

So in the case of your example:

$\displaystyle 1002^{3755}\equiv 2^{3755} \pmod{10}$

Now (you should know this) $\displaystyle 2^{10}=1024$
so:

$\displaystyle 2^{3755}=(2^{10})^{375}2^5\equiv 4^{375} \times 2 \pmod{10}$

$\displaystyle 4^{375} \times 2=2^{751}=(2^{10})^{75} \times 2 \equiv 4^{75} \times 2 \pmod{10}$

$\displaystyle 4^{75} \times 2=2^{151}=(2^{10})^{15} \times 2 \equiv 4^{15} \times 2 \pmod{10}$

$\displaystyle 4^{15} \times 2=2^{31}=(2^{10})^3 \times 2\equiv 4^3 \times 2 \pmod{10}$

and:

$\displaystyle 4^3 \times 2=128 \equiv 8 \pmod{10}$

Hence:

$\displaystyle 1002^{3755}\equiv 8 \pmod{10}$

and so the least significant digit of $\displaystyle 1002^{3755}$ is $\displaystyle 8$.

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