> 把自然数N分解成若干个互不相同的正整数，使乘积最大；
> 
> 由于这种分解的数目是有限的，所以最大积存在；
> 
> 假设最大积的分解为
> 
> n=a1+a2+a3+...+a[t-2]+a[t-1]+a[t]
> 
> (a1<a2<a3<...<a[t-2]<a[t-1]<a[t])
> 
> 我们来证明这个数列的一些性质；
> 
> 1.1<a1
> 
> if a1=1, then a1(=1), a[t] together could be replaced by a[t]+1.
> 
> 2.to all i, a[i+1]-a[i]<=2;
> 
> if some i make a[i+1]-a[i]>=3,
> 
> then a[i],a[i+1] together could be replaced by a[i]+1,a[i+1]-1 together.
> 
> 3. at MOST one i, fits a[i+1]-a[i]=2
> 
> if i<j and a[i+1]-a[i]=2 and a[j+1]-a[j]=2 then
> 
> a[i],a[j+1] could be replaced by a[i]+1, a[j+1]-1
> 
>  
> 
> 4. a1<=3
> 
> if a1>=4, then a1,a2 together could be replaced by 2, a1-1, a2-1 together 
> 
> 5. if a1=3 and one i fits a[i+1]-a[i]=2 then i must be t-1
> 
> if i<t-1 then a[i+2] could be replaced by 2, a[i+2]-2 together
> 
> Now, from the five rules above, we could make the mutiple maximum.
> 
> to an N, find the integer k, fits
> 
> A=2+3+4+...+(k-1)+k <= N < A+(k+1)=B
> 
> Suppose N = A + p, (0 <= p < k+1)
> 
> 1) p=0, then answer is Set A
> 
> 2) 1<=p<=k-1 then answer is Set B - { k+1-p }
> 
> 3) p=k, then answer is Set A - {2} + {k+2}
> 
> We can prove this is the best choice with ease,
> 
> as any other choice will lead to at least one of the following:
> 
> 1) a1>=4 or a1=1
> 
> 2) two a[i+1]-a[i]=1 or one a[i+1]-a[i]=2
> 
> 3) a1=3 and some i<t-2 fits a[i+1]-a[i]=1