The reminder, when $4^{87}+6^{87}$ is divided by 25 is 

$4^{87}+6^{87}$

${\left(5-1\right)}^{87}+{\left(5+1\right)}^{87}$

${\left(1+5\right)}^{87}-{\left(1-5\right)}^{87}$

$2\left[{ }^{87}{C}_1.5{+}^{87}{C}_3{.5}^3+{\mathrm{.....}}^{87}{C}_{87}{.5}^{87}\right]$

$10.87+2\left[{ }^{87}{C}_3{.5}^3+{\mathrm{......}}^{87}{C}_{87}{.5}^{87}\right]$

=870+an expression divisible by 25

 $\Rightarrow \frac{870}{25}=34+\frac{20}{25}$  Reminder is 20