It is known that "if three prime numbers a, B and C greater than 3 satisfy the relation 2A + 5B = C, then a + B + C is a multiple of integer n". What is the maximum possible value of integer n in this theorem? Please prove your conclusion

It is known that "if three prime numbers a, B and C greater than 3 satisfy the relation 2A + 5B = C, then a + B + C is a multiple of integer n". What is the maximum possible value of integer n in this theorem? Please prove your conclusion

This paper proves that: a + A + B + B + C = a + B + B + C = a + B + B + 2A + 5B = 3 (a + 2b), and obviously, 3|a + B + C, if the remainder of a and B after 3 is respectively RA and Rb, then RA ≠0, and Rb ≠ 0. If RA ≠ Rb, then RA = 2, Rb = 1, or RA = 1, or RA = 1, Rb = 2, then 2A + 5B = 2 (3m + 2) + 5 (3 (3m + 2) + 5 (3N + 5 (3N + 1) = 3 (2m + 5N + 5N + 3) (2m + 5N + 2m + 5N + 3) or 2A + 5B = 2 (2 (2) (3P + 2 (3P + 1) + + 1) + 2) (2 (3 (3P + 1) + 5 (3P) + 5) + 5) (5 (3P = 2) (2P only RA = RA = RA = RA = RA = 3) then ra= Then a + 2B must be a multiple of 3, so a + B + C is a multiple of 9. A and B are prime numbers greater than 3. According to the meaning of the topic, take a = 11, B = 5, then 2A + 5B = 2 × 11 + 5 × 5 = 47, a + B + C = 11 + 5 + 47 = 63, take a = 13, B = 7, then 2A + 5B = 2 × 13 + 5 × 7 = 61, a + B + C = 13 + 7 + 61 = 81, and (63, 81) = 9, so 9 is the maximum possible value