This problem concerns languages over the alphabet Σ = {1}. For any two integers q, r ≥0, define the language Lr, q over Σ as
Lr, q ={1mq+r : m ≥0}={1q,1q+r,12q+r,...}.
For instance, L1,3 ={1,1111,1111111,...}.
(a) Show that for every q and r, the language Lr, q is regular.
(b) Show that if L is a regular language over Σ, then L can be written as
L = Lr1,q ∪ Lr2,q ∪···∪ Lrk, q ∪ L0,
where 0 ≤ r1 < ··· < rk are integers and L0 is a finite set.

Write a program to analyze student’s scores. each student’s id and 4 original test scores are saved in the attached file “student_id_scores.txt”. assume there are exactly 15 students in the class.• the program first opens the file, reads student’s ids and scores from the file and saves them in two arrays: o create a one-dimensional string array “ids” to store the student’s ids.o create a (parallel) two-dimensional int array “scores” to store student’s original test scores. this array has 5 columns, the original 4 test scores should be saved in the first 4 columns, and the last column will be used to save the average score after the calculation in next step.• the program then retrieves test scores from the array “scores”, calculates average score and determines the final letter grade based on the rules in the table below for each student. assume average score is integer value.average score = (score1 + score2 + score3 + score4) / 4average score letter grade90 ~ 100 a80 ~ 89 b70 ~ 79 c60 ~ 70 d0 ~ 60 f• the average score should be saved back in the 5th column of “scores” array.• create a new parallel one-dimensional char array “grades” to save each student’s letter grade.• program also needs to find the number of a, number of b, number of c, number of d and number of f grade. as well as the highest, lowest and average score of the class.• finally, program will display all of the original scores and all results on the computer screen using the format shown as the sample output below.• the program must contain at least the following functions: (1) a function “getscores” to read and store data into “ids” and “scores” arrays.(2) a function “analyzescores” to calculate student’s average score, determine letter grade, get the number of a, b, c, d and f, find the highest, lowest and average score of the class.(3) a function “displayresults” to display all of the results