$[$ Arithmetic of residues (other). ]

Find the remainder of the division of $2^{100}$ by 3.
$2^{100}=4^{50} \equiv 1^{50}=1(\bmod 3)$.

## Answer

1. 

In a class of 30 people, can it be that 9 of them have 3 friends (in this class), 11 have 4 friends, and 10 have 5 friends?

## Solution

In the corresponding graph, there would be 30 vertices, 9 of which would have a degree of 3, 11 a degree of 4, and 10 a degree of 5. However, such a graph would have 19 vertices with odd degrees, which contradicts problem 87972 b).

## Answer

It cannot.