<s>[INST] How many numbers from $1$ to $1000$ can be written as the sum of $4$s and $5$s? 
I like the numbers $4$ and $5$. I also like any number that can be added together using $4$s and $5$s. Eg,
  $$9 = 4+5 \qquad 40 = 5 + 5 + 5 + 5 + 5 + 5 + 5 +5$$ 
  How many number have this property from 1 to 1000?

Multiples of $4$s and $5$s are easy, but how do I calculate the number of numbers from different combinations of adding $4$ and $5$? (And which ones are different from multiples of $4$ and $5$?)
 [/INST] Google frobenius coin problem.
But notice.  If $K = 4m + 5n$ is such number then $K+4 = 4(m+1)+5n$ is such a number and all $K + 4a = 4(m+a) + 5n$ will be such numbers.
$12 = 3*4$ and $13=2*4 + 5$ and $14=4+2*5$ and $15=3*5$.  So every $12 + 4a$ and $13+4b$ and $14+4b$ and $15+4c$ will be such numbers  And that is every number greater than or equal to $12$.
The hard part is finding out that $11$ is the largest that can't be done.
ANd then counting that the ones that are less than $11$ than can be done are $4; 8=2*4; 5;9=5+4;10=2*5$ and all the others $1,2,3,6,7,11$ cant be done.  
So that is $6$ that can't be done and all the rest that can. So $994$.
.....
Another way.... harder, but more intuitive for me....
If $N = 4k + r$ where $r=0,1,2,3$ is the remainder.  We can do $N = 4k+r = 4k-4r + 5r = 4(k-r)+5r$ provided that $k \ge r$.
So if $k=0$ then we can't do this.  If $k=1$ then if $r \le 1$, i.e. if $N=4*1+0 =4$ or $N = 4*1 + 1 = 0*1 + 5$, we can do this. but we can't do this if $k=1$ and $r=2,3$ i.e. if $N =4*1 +2=6$ and $N = 4*1 + 3 = 7$.
In $k=2$ and $r\le 2$ we can do this.  $N=4*2+ 0 =8; N=4*2 + 1=4*1 + 5 = 9$; and $N=4*2 +2 = 4*0 + 2*5$.  But if $r=3$ we can not; $N=4*2+3=4*1 +5*1+2 = 4*0 +5*2+1$ can not be done.
But if $k \ge 3$ and $r \le 3$ we can do it and that is the case for all $N \ge 12$.
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