swerve wrote:Find the units digit of 556^{17n}+339^{5m+15n} , where m and n are positive integers.
(1) 14m + 12n = 360.
(2) n is the smallest 2-digit positive integer divisible by 5.
Source: e-GMAT
$$\left\langle M \right\rangle \,\, = \,\,{\rm{units}}\,\,{\rm{digit}}\,\,{\rm{of}}\,\,M$$
$$\left\langle {{{556}^n}} \right\rangle = \left\langle {{6^n}} \right\rangle = 6\,\,,\,\,\forall n \ge 1\,\,{\mathop{\rm int}} $$
$$\left\langle {{{339}^k}} \right\rangle = \left\langle {{9^k}} \right\rangle = \left\{ \matrix{
\,\,9\,\,,\,\,\forall k \ge 1\,\,{\rm{odd}} \hfill \cr
\,\,1\,\,,\,\,\forall k \ge 2\,\,{\rm{even}} \hfill \cr} \right.\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{{339}^{5\left( {m + 3n} \right)}}} \right\rangle = \left\{ \matrix{
\,\,9\,\,,\,\,\forall \,m + 3n\,\, \ge \,\,5\,\,{\rm{odd}} \hfill \cr
\,\,1\,\,,\,\,\forall \,m + 3n\,\, \ge \,\,4\,\,{\rm{even}} \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\,\left( {m,n\,\, \ge 1\,\,{\rm{ints}}} \right)$$
$$\left\langle {{{556}^{17n}} + {{339}^{5\left( {m + 3n} \right)}}} \right\rangle \,\,\, = \,\,\,\left\{ \matrix{
\,\,\left\langle {6 + 9} \right\rangle = 5\,\,,\,\,\forall \,m + 3n\,\, \ge \,\,5\,\,{\rm{odd}} \hfill \cr
\,\,\left\langle {6 + 1} \right\rangle = 7\,\,\,,\,\,\forall \,m + 3n\,\, \ge \,\,4\,\,{\rm{even}} \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\,\left( {m,n\,\, \ge 1\,\,{\rm{ints}}} \right)$$
$$?\,\, = \left\langle {{{556}^{17n}} + {{339}^{5\left( {m + 3n} \right)}}} \right\rangle \,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,?\,\,\,:\,\,\,m + 3n\,\,\,{\rm{odd/even}}\,\,\,\,\,\,\,\,\,\,\left[ {\,m,n\,\, \ge 1\,\,{\rm{ints}}\,} \right]\,$$
$$\left( 1 \right)\,\,\,4m + 12n = 360\,\,\,\,\, \Rightarrow \,\,\,\,m + 3n = 90\,\,\left( {{\rm{even}}} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.\,\,\,\,\,\,\,\left( {? = 7} \right)$$
$$\left( 2 \right)\,\,n = 10\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,m = 1\,\,\,\, \Rightarrow \,\,\,\,m + 3n = 31\,\,\left( {{\rm{odd}}} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 5\,\, \hfill \cr
\,{\rm{Take}}\,\,m = 2\,\,\,\, \Rightarrow \,\,\,\,m + 3n = 32\,\,\left( {{\rm{even}}} \right)\,\,\,\, \Rightarrow \,\,\,\,\,? = 7\,\, \hfill \cr} \right.$$
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
