By the sounds of it, you are saying this:
I have a linear homogeneous equation f(a,b,c,...,n)=0, and I have some monomial like b, ab^2, or cd^2m^5 and I want to divide out by that coefficient to make this term have coefficient 1.
The quickest way to do this might be:
EXP = a + 5 b + 10 c;
This will divide out by the coefficient of
b in the expression
EXP. You can do this with other variables in the expression:
MONIC[EXP_, n_] := Expand[EXP/Coefficient[EXP, n]]
However, if you try to do something more complicated:
EXP = a + 5 b + 10 c + 3 a b^2 + 11 c d m^5;
MONIC[EXP, a b^2]
MONIC[EXP, c d m^5]
it will fail to work because it doesn't treat nonlinear expressions properly. To fix that, you can use the whole list of monomials:
MONIC2[EXP_, n_] :=
Expand[EXP/Select[MonomialList[EXP]/n, IntegerQ[#] &][]]
MONIC2[EXP, a b^2]
MONIC2[EXP, c d m^5]
That will do the trick. If you want to use this in a fool-proof way, you'd need to include some safeguards to prevent division by zero and/or picking from a list that has no elements. But assuming the user inputs something that ought to have an answer, this will give it. Issues could also occur trying to use symbols that have already been assigned values of some kind.
Your equation will now be
And as others have noted, probably good to stick to lowercase letters for variables.