10、If $\frac{2^{12}+1}{2^{4}+1}=2^{a_{1}}+2^{a_{2}}+\ldots+2^{a_{k}}$ where $a_{1} \sim a_{\mathrm{k}}$ are natural numbers, and $a_{1}<a_{2}<\cdots$ $<a_{2}$, then $\mathrm{k}=$ $\qquad$
【Answer】 5
【Analysis】 Method 1: Binary. $2^{a_{1}}+2^{a_{2}}+\ldots+2^{a_{k}}$ can be converted to binary, $\frac{2^{12}+1}{2^{4}+1}=\frac{4097}{17}=241 ;: 241$ converted to binary, short division by 2, take the remainder in reverse, (11110001) $2=2^{8}+2^{7}+2^{6}+2^{5}+2^{0}$, a total of 5 numbers;

Method 2: In fact, this problem can be more simply solved using enumeration, employing a greedy algorithm (simply put, finding the largest power of 2 to subtract). Operations:
$$
241-128=113,113-64=49,49-32=17,17-16=1 ; \quad 241=128+64+32+16+1
$$