Remainder when divided by 15

$\begin{array}{rcl} &7^{23}+8^{79}+11^{105}\\ \equiv&7+8+11\\ \equiv&2 \end{array}\\\mbox{(mod 3)}$

$\begin{array}{rcl} &7^{23}+8^{79}+11^{105}\\ \equiv&7^3+8^3+11\\ \equiv&2^3+3^3+1\\ \equiv&8+27+1\\ \equiv&1 \end{array}\\\mbox{(mod 5)}$

$\therefore7^{23}+8^{79}+11^{105}\equiv2\mbox{(mod 5)}$

$7^{23}\equiv 7\times 7^{22}\equiv 7\times 49^{11}\equiv 7\times 4^{11}\equiv 7\times 4 \times 4^{10}\equiv 28\times 16^5$ $\equiv 28\times 1^{5}\equiv 28\equiv 13 \space (\bmod 15)$ $8^{79}\equiv 8\times 8^{78}\equiv 8\times 64^{39}\equiv 8\times 4^{39}\equiv 32\times 4^{38}\equiv 2\times 16^{19}$ $\equiv 2\times 1^{19}\equiv 2\space (\bmod 15)$ $11^{105}\equiv 11\times 11^{104}\equiv 11\times 121^{52}\equiv 11\times 1^{52}\equiv 11\space (\bmod 15)$ $\Rightarrow 7^{23}+8^{79}+11^{105}\equiv \red{13+2}+11\equiv 11 (\bmod 15)$