Suppose there are 100 lockers in a school hallway and they are all closed. Only 25 students come through the hallway and start opening and closing the lockers. The first student opened
all the lockers. The second student closed every second locker and so forth. The Third student changes the state
of every 3rd locker. The fourth student changes the state of every 4th locker, the fifth student changes the state
of every 5th locker and so on, until the 25th student.
1. What is the state of the following locker numbers: #23, #55, #81, #100?

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!

15 units use socratic

step-by-step explanation: 15 units

figure c

step-by-step explanation:

hope this .