pri187 wrote:
Hi Anurag,
Im a little confused here.
123! is a multiple of 4, agreed.
So is 3 or any number less then or equal to 123 will be a multiple of 123!
if i consider 123! = 3n , then power of 2 is multiple of 3, so according to cyclicity of 2 the unit digit will be 8.
Please let know where am I going wrong
Hi, pri187.
I will take the liberty to answer your question, simply because I don´t have any clue whether Anurag is still active (at least in this forum). (*)
Let < N > denote the unit´s digit of N, for instance: <2> = 2 and <2^4> = 6 (because 2^4 = 16). With this notation in mind, please note that:
$$\left\langle {{2^{12}}} \right\rangle = \left\{ \matrix{
\left\langle {{{\left( {{2^3}} \right)}^4}} \right\rangle = \left\langle {{8^4}} \right\rangle = 6 \hfill \cr
\left\langle {{{\left( {{2^4}} \right)}^3}} \right\rangle = \left\langle {{6^3}} \right\rangle = 6 \hfill \cr} \right.$$
In other words, it´s not just a matter of divisibility (by 3 or by 4), but also the recognition that once we get the unit´s digit equal to 6, any "sequential" positive integer power will keep this last digit 6...
In our case, we may proceed as follows:
$$\left\langle {{2^{123!}}} \right\rangle = \left\{ \matrix{
\left\langle {{{\left( {{2^4}} \right)}^{\,{{123!} \over 4}}}} \right\rangle = \left\langle {{6^{{{123!} \over 4}}}} \right\rangle = 6 \hfill \cr
\left\langle {{{\left( {{2^3}} \right)}^{\,{{123!} \over 3}}}} \right\rangle = \left\langle {{8^{{{123!} \over 3}}}} \right\rangle = \left\langle {{{\left( {{8^4}} \right)}^{{{123!} \over {3 \cdot 4}}}}} \right\rangle = \left\langle {{6^{{{123!} \over {3 \cdot 4}}}}} \right\rangle = 6 \hfill \cr} \right.$$
I hope things are clearer now!
Regards,
Fabio.
(*) P.S.: in approximately 2012 I exchanged here many interesting posts with him, through which I feel honored to consider myself his friend since then.
(He is an outstanding expert and an even more impressive soul, I must say. His intellectual power never crossed his friendly nature. No ego at all.)
If someone here has any contact with him, please send to him my best wishes!
