已知 $2^{1003} + 2^{1000} = 2^n \times 3^m$，求 $m$ 和 $n$ 的值。 答：m = ___， n = ___

1. Factorise the common power of 2 from the left-hand side.

$2^{1003}+2^{1000}=2^{1000}(2^3+1)$

2. Simplify the bracket to express it as a power of 3.

$2^3+1=8+1=9=3^2$

So,
$2^{1000}(2^3+1)=2^{1000}\times 3^2.$

3. Compare with the given form $2^n\times 3^m$ to identify $m$ and $n$.

Since
$2^{1003}+2^{1000}=2^{1000}\times 3^2,$
comparing with
$2^n\times 3^m,$
we get
$n=1000,\quad m=2.$