5. Let real numbers $a, b$ satisfy $3^{a}+13^{b}=17^{a}, 5^{a}+7^{b}=11^{b}$, then the relationship between $a, b$ is
A. $a>b$
B. $a=b$
C. $a<b$
D. Uncertain
If $a \geqslant b, 17^{a}=3^{a}+13^{b} \leqslant 3^{a}+13^{a} \Rightarrow\left(\frac{3}{17}\right)^{a}+\left(\frac{13}{17}\right)^{a} \geqslant 1>\frac{3}{17}+\frac{13}{17}$,
combining the monotonicity of the function $y=\left(\frac{3}{17}\right)^{x}+\left(\frac{13}{17}\right)^{x}$ we get $a1$, which is a contradiction. Therefore, the answer is $C$.