Question

so that the cube of any positive integer is of the form 4m,4m+1 or 4m+3, for some integer m

Answer 1

using Euclid division lemme , we get
a=b×q+r
b=4
q is any positive integer
r= 0,1,2,3
a=4q+0

let sq both sides
a^2=16q^2
a^2=4(4q^2)
a^2=4m

hence this method cam solve rest of this

Answer 2

take all 4 cases:
I) a = 4q + 0 = 4q
II) a = 4q + 1
III) a = 4q + 2
IV) a = 4q + 3