# Need help with this congruency problem

• Dec 13th 2009, 11:03 AM
steph3824
Need help with this congruency problem
Show (23^(37)+15^(16)) is congurent to 5 (mod 18).

Not sure how to do this need some help please!!
• Dec 13th 2009, 12:46 PM
tonio
Quote:

Originally Posted by steph3824
Show (23^(37)+15^(16)) is congurent to 5 (mod 18).

Not sure how to do this need some help please!!

I did it two ways and I get something different. The following is done modulo 18:

$23^{37}+15^{16}=5^{37}+5^{16}\cdot 3^{16}=5^{16}\left(5^{21}+3^{16}\right)=\left(5^3\ right)^5\cdot 5\left(\left(5^3\right)^5+\left(3^4\right)^4\right )$ $=(-1)^5\cdot 5\left((-1)^5+9\right)=(-5)\left((-1)+9\right)=(-5)\cdot 8=-40=-4=14\!\!\!\!\pmod{18}$
Of course, I could be wrong...

Tonio
• Dec 13th 2009, 03:03 PM
aidan
Quote:

Originally Posted by steph3824
Show (23^(37)+15^(16)) is congurent to 5 (mod 18).

Not sure how to do this need some help please!!

$23^{37} \equiv 5 \mod 18$

$15^{16} \equiv 9 \mod 18$

& $5+9 \equiv 14 \mod 18$

That agrees with tonio.