Help:=proc(): print(` `): end:

# Part 1 ----------------------------------------------
Fc:=proc(n,K) local B, b, A, i:
if n=1 then
  return(1):
fi:

with(linalg):
  B:=matrix(2,2,[[1,1],[1,0]]):

# A(n) = [F(n),F(n-1)]  A(1) = [1,0]
b:= [1,0]:

for i from 1 to n-1 do
  A:=multiply(B,b):
  A:=op(convert(matrix(1,2,op(A)),list)):

  A:= A mod K:
  b:=A:
od:

op(A)[1]:
end:


# Part 2 ----------------------------------------------
Tc:=proc(n,K) local B, b, A, i:
if n<=2 then
  return(1):
elif n=3 then
  return(2):
fi:


with(linalg):
  B:=matrix(3,3,[[1,1,1],[1,0,0],[0,1,0]]):

# A(n) = [F(n),F(n-1),F(n-2)]  A(1) = [1,0]
b:= [2,1,1]:

for i from 1 to n-1 do
  A:=multiply(B,b):
  A:=convert(A,list):

  A:= A mod K:
  b:=A:
od:

op(A)[1]:
end: