Force required to keep block from sliding down (check)

1. The problem statement, all variables and given/known data
The two blocks (with m=16kg and M=88kg) shown in Fig. 6-56 are not attached. The coefficient of static friction between the blocks is [itex]\mu[/itex]^stat=0.38, but the surface beneath the larger block is frictionless. what is the minimum magnitude of the horizontal force [itex]\vec{F}[/itex] required to keep the smaller block from slipping down the larger block?

Fig. 6-56 looks approximately like this.

....______ ________
....|.......||...........|
[itex]\vec{F}[/itex].|..m...||...........|
....|_____||.....M...|
..............|...........|
..............|_______|
^{_________________________________}
FRICTIONLESS GROUND

2. Relevant equations
[itex]\sum[/itex][itex]\vec{F}[/itex]=m[itex]\vec{a}[/itex]

[itex]\vec{F}[/itex]_fric=[itex]\mu[/itex]^stat[itex]\vec{N}[/itex]


3. The attempt at a solution
I denote [itex]\vec{F}[/itex] from the problem text as [itex]\vec{F}[/itex]_A

I denote the force of block m and block M, [itex]\vec{F}[/itex]_mM

Likewise, the force of block M and block m is denoted [itex]\vec{F}[/itex]_Mm. I treat this as the "normal" force for the friction equation, but I'm not sure if this is correct.

F_gravm=mg=156.8 N

[itex]\vec{a}[/itex]=[itex]\vec{F}[/itex]_A/104


[itex]\vec{F}[/itex]_mM=[itex]\vec{F}[/itex]_Mm=[itex]\frac{88}{104}[/itex][itex]\vec{F}[/itex]_A

[itex]\vec{F}[/itex]_fric=[itex]\mu[/itex][itex]\vec{F}[/itex]_Mm=.38*[itex]\frac{88}{104}[/itex][itex]\vec{F}[/itex]_A=.322[itex]\vec{F}[/itex]_A

To prevent downward motion [itex]\vec{F}[/itex]_fric must cancel out (i.e. be at least) [itex]\vec{F}[/itex]gravm

[itex]\vec{F}[/itex]_fric=[itex]\vec{F}[/itex]gravm
.322[itex]\vec{F}[/itex]_A=156.8N
[itex]\vec{F}[/itex]_A=487.7 N

Final answer 487.7 N

Thanks