S=1 square-2 square+3 square-4 square+5 square -…… +99 Square-100 square +101 square, find the remainder of S divided by 103

S=1 square-2 square+3 square-4 square+5 square -…… +99 Square-100 square +101 square, find the remainder of S divided by 103

S=1 square-2 square+3 square-4 square+5 square -…… +99 Square-100 square +101 square
=1+(3 Square-2 square)+(4 square-3 square)+...+(101 square-100 square)
=1+(3-2)(3+2)+(4-3)(4+3)+...+(101-100)(101+100)
=1+2+3+...+101
=(1+101)101/2
=5151
5151/103=50+1
So the remainder of S divided by 103 is 1

1+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+

Let a=1+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2
Then 2a =2+2, the third power of the second power +2, the fourth power of the fourth power +.+2, the ninety-ninth power +2, the one hundredth power
Subtract,2a-a=a on the left.
Right, same offset in the middle.
Then a =1 to the 100th power of 2
So 1+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2