From the closure of Z, we know that A is an arbitrary integer, IM (φ) belongs to Z, φ (x+y) = φ (x)+φ (y) always holds and always belongs to Z, so if A is an integer, it is an endomorphism. If a is not an integer, φ (1) = A does not belong to z, so it is not endomorphism.

Obviously, if a=0 and φ (x) equals 0, then IM (φ) = 0 is a simple endomorphism.

A complete endomorphism requires that for all integers Y, X exists such that φ (X) = Y, that is, IM (φ) = Z, so a is less than or equal to 1, A = 1/k, and k belongs to Z.

For example, if a=2 and y= 1, then there is no X belonging to Z, so 2x=y= 1, because x=0.5 does not belong to Z. Therefore, a=2 does not satisfy complete endomorphism.

The last question is a = 1 or-1.