pigeon hole principle :
prove that given any set of n + 1 integers, there must be at least one pair among them whose difference is divisible by n

A proof can be as follows:

Step-by-step explanation:

Let be a set of integers. By the division algorithm the possible remainders when we divide by are . Then, each integer can be written as:

Observe that the set of remainders has elements and each element has possible values. By the Pigenhole principle at least two remainders have the same value. Suppose that this two elements are . Then,

Where . Then,  . Then we have that divides .

34.5 cm

step-by-step explanation:

generally for a circular segment of angle, θ° and radius r

arc length

= (θ/360) x circumference

= (θ/360) x 2πr

in our case, θ = 60°, π=3.142 and r = 33 cm

substitute into above formula

arc length

= (60/360) x 2(3.142)( 33) = 34.5 cm