let

k = the number of messages Kesha sent

j = the number of messages John sent

m = the number of messages Milan sent

where j, k and m are non-negative integers

Kesha, John, and Milan sent a total of 113 text messages for the weekend.

k + j + m = 113

Kesha sent 4 times as many messages as john.

k = 4j

Milan sent 5 times as many messages then John.

m = 5j

by solving the system of equations

k + j + m = 113

k = 4j

m = 5j

we find

k = 45.2

j = 11.3

m = 56.6

click here to see the step by step solution of the system of equations

Click to see all the steps

the solutions of the system are not integers thus the problem has no solution.