Sports games

Pupils of same school participated district sports games. When dividing into teams found that in the case of the creation teams with 4 pupils remaining 1 pupil, in the case of a five-member teams remaining 2 pupils and in the case of six-members teams remaining 3 pupils. How many pupils in this school attended games if one school could participated a maximum of 80 pupils?

Correct result:

n =  57

Solution:

0 = 4×0 + 0 = 5×0 + 0 = 6×0 + 0
1 = 4×0 + 1 = 5×0 + 1 = 6×0 + 1
2 = 4×0 + 2 = 5×0 + 2 = 6×0 + 2
3 = 4×0 + 3 = 5×0 + 3 = 6×0 + 3
4 = 4×1 + 0 = 5×0 + 4 = 6×0 + 4
5 = 4×1 + 1 = 5×1 + 0 = 6×0 + 5
6 = 4×1 + 2 = 5×1 + 1 = 6×1 + 0
7 = 4×1 + 3 = 5×1 + 2 = 6×1 + 1
8 = 4×2 + 0 = 5×1 + 3 = 6×1 + 2
9 = 4×2 + 1 = 5×1 + 4 = 6×1 + 3
10 = 4×2 + 2 = 5×2 + 0 = 6×1 + 4
11 = 4×2 + 3 = 5×2 + 1 = 6×1 + 5
12 = 4×3 + 0 = 5×2 + 2 = 6×2 + 0
13 = 4×3 + 1 = 5×2 + 3 = 6×2 + 1
14 = 4×3 + 2 = 5×2 + 4 = 6×2 + 2
15 = 4×3 + 3 = 5×3 + 0 = 6×2 + 3
16 = 4×4 + 0 = 5×3 + 1 = 6×2 + 4
17 = 4×4 + 1 = 5×3 + 2 = 6×2 + 5
18 = 4×4 + 2 = 5×3 + 3 = 6×3 + 0
19 = 4×4 + 3 = 5×3 + 4 = 6×3 + 1
20 = 4×5 + 0 = 5×4 + 0 = 6×3 + 2
21 = 4×5 + 1 = 5×4 + 1 = 6×3 + 3
22 = 4×5 + 2 = 5×4 + 2 = 6×3 + 4
23 = 4×5 + 3 = 5×4 + 3 = 6×3 + 5
24 = 4×6 + 0 = 5×4 + 4 = 6×4 + 0
25 = 4×6 + 1 = 5×5 + 0 = 6×4 + 1
26 = 4×6 + 2 = 5×5 + 1 = 6×4 + 2
27 = 4×6 + 3 = 5×5 + 2 = 6×4 + 3
28 = 4×7 + 0 = 5×5 + 3 = 6×4 + 4
29 = 4×7 + 1 = 5×5 + 4 = 6×4 + 5
30 = 4×7 + 2 = 5×6 + 0 = 6×5 + 0
31 = 4×7 + 3 = 5×6 + 1 = 6×5 + 1
32 = 4×8 + 0 = 5×6 + 2 = 6×5 + 2
33 = 4×8 + 1 = 5×6 + 3 = 6×5 + 3
34 = 4×8 + 2 = 5×6 + 4 = 6×5 + 4
35 = 4×8 + 3 = 5×7 + 0 = 6×5 + 5
36 = 4×9 + 0 = 5×7 + 1 = 6×6 + 0
37 = 4×9 + 1 = 5×7 + 2 = 6×6 + 1
38 = 4×9 + 2 = 5×7 + 3 = 6×6 + 2
39 = 4×9 + 3 = 5×7 + 4 = 6×6 + 3
40 = 4×10 + 0 = 5×8 + 0 = 6×6 + 4
41 = 4×10 + 1 = 5×8 + 1 = 6×6 + 5
42 = 4×10 + 2 = 5×8 + 2 = 6×7 + 0
43 = 4×10 + 3 = 5×8 + 3 = 6×7 + 1
44 = 4×11 + 0 = 5×8 + 4 = 6×7 + 2
45 = 4×11 + 1 = 5×9 + 0 = 6×7 + 3
46 = 4×11 + 2 = 5×9 + 1 = 6×7 + 4
47 = 4×11 + 3 = 5×9 + 2 = 6×7 + 5
48 = 4×12 + 0 = 5×9 + 3 = 6×8 + 0
49 = 4×12 + 1 = 5×9 + 4 = 6×8 + 1
50 = 4×12 + 2 = 5×10 + 0 = 6×8 + 2
51 = 4×12 + 3 = 5×10 + 1 = 6×8 + 3
52 = 4×13 + 0 = 5×10 + 2 = 6×8 + 4
53 = 4×13 + 1 = 5×10 + 3 = 6×8 + 5
54 = 4×13 + 2 = 5×10 + 4 = 6×9 + 0
55 = 4×13 + 3 = 5×11 + 0 = 6×9 + 1
56 = 4×14 + 0 = 5×11 + 1 = 6×9 + 2
57 = 4×14 + 1 = 5×11 + 2 = 6×9 + 3

$n = 4x+1 = 5y+2 = 6z+3 \ \\ x,y,z \in N \ \\ n \le 80 \ \\ n = 57$

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