Question 41506
A student has noticed that every squared natural number is either a multiple of 4 or one larger than a multiple of 4, but isn't sure if this is always tru. Use algebraic reasoning to explain to the student why this is always tru.
natural number could be 
case 1...even....=2m
its square=2m*2m=4m^2...divisible by 4
case 2....odd.....2m-1
its square =(2m-1)^2=4m^2-4m+1=4(m^2-m)+1=4k+1...where k is an integer.
hence square of every natural number is either a multiple of 4 or leaves a remainder of 1 when divided by 4.