Để chứng minh rằng tồn tại số có dạng 555...5 (gồm toàn chữ số 5) chia hết cho 13, ta sẽ sử dụng một số lý thuyết về số học và tính chất của số chia hết.

Giả sử số có dạng 555...5 gồm \( n \) chữ số 5. Ta có thể biểu diễn số này dưới dạng:
\[ N = 5 \times \frac{10^n - 1}{9} \]

Chúng ta cần chứng minh rằng tồn tại \( n \) sao cho \( N \) chia hết cho 13, tức là:
\[ 5 \times \frac{10^n - 1}{9} \equiv 0 \pmod{13} \]

Điều này tương đương với:
\[ \frac{10^n - 1}{9} \equiv 0 \pmod{\frac{13}{\gcd(13, 5)}} \]

Vì 13 và 5 là nguyên tố cùng nhau (gcd(13, 5) = 1), ta có:
\[ \frac{10^n - 1}{9} \equiv 0 \pmod{13} \]

Do đó, ta cần:
\[ 10^n - 1 \equiv 0 \pmod{13} \]

Điều này có nghĩa là:
\[ 10^n \equiv 1 \pmod{13} \]

Bây giờ, ta cần tìm \( n \) sao cho \( 10^n \equiv 1 \pmod{13} \). Để làm điều này, ta sẽ kiểm tra các giá trị của \( n \):

- \( 10^1 \equiv 10 \pmod{13} \)
- \( 10^2 \equiv 100 \equiv 9 \pmod{13} \)
- \( 10^3 \equiv 10 \times 9 \equiv 90 \equiv -4 \pmod{13} \)
- \( 10^4 \equiv -4 \times 10 \equiv -40 \equiv 6 \pmod{13} \)
- \( 10^5 \equiv 6 \times 10 \equiv 60 \equiv 8 \pmod{13} \)
- \( 10^6 \equiv 8 \times 10 \equiv 80 \equiv 2 \pmod{13} \)
- \( 10^7 \equiv 2 \times 10 \equiv 20 \equiv 7 \pmod{13} \)
- \( 10^8 \equiv 7 \times 10 \equiv 70 \equiv 5 \pmod{13} \)
- \( 10^9 \equiv 5 \times 10 \equiv 50 \equiv 11 \pmod{13} \)
- \( 10^{10} \equiv 11 \times 10 \equiv 110 \equiv 6 \pmod{13} \)
- \( 10^{11} \equiv 6 \times 10 \equiv 60 \equiv 8 \pmod{13} \)
- \( 10^{12} \equiv 8 \times 10 \equiv 80 \equiv 2 \pmod{13} \)
- \( 10^{13} \equiv 2 \times 10 \equiv 20 \equiv 7 \pmod{13} \)
- \( 10^{14} \equiv 7 \times 10 \equiv 70 \equiv 5 \pmod{13} \)
- \( 10^{15} \equiv 5 \times 10 \equiv 50 \equiv 11 \pmod{13} \)
- \( 10^{16} \equiv 11 \times 10 \equiv 110 \equiv 6 \pmod{13} \)
- \( 10^{17} \equiv 6 \times 10 \equiv 60 \equiv 8 \pmod{13} \)
- \( 10^{18} \equiv 8 \times 10 \equiv 80 \equiv 2 \pmod{13} \)
- \( 10^{19} \equiv 2 \times 10 \equiv 20 \equiv 7 \pmod{13} \)
- \( 10^{20} \equiv 7 \times 10 \equiv 70 \equiv 5 \pmod{13} \)
- \( 10^{21} \equiv 5 \times 10 \equiv 50 \equiv 11 \pmod{13} \)
- \( 10^{22} \equiv 11 \times 10 \equiv 110 \equiv 6 \pmod{13} \)
- \( 10^{23} \equiv 6 \times 10 \equiv 60 \equiv 8 \pmod{13} \)
- \( 10^{24} \equiv 8 \times 10 \equiv 80 \equiv 2 \pmod{13} \)
- \( 10^{25} \equiv 2 \times 10 \equiv 20 \equiv 7 \pmod{13} \)
- \( 10^{26} \equiv 7 \times 10 \equiv 70 \equiv 5 \pmod{13} \)
- \( 10^{27} \equiv 5 \times 10 \equiv 50 \equiv 11 \pmod{13} \)
- \( 10^{28} \equiv 11 \times 10 \equiv 110 \equiv 6 \pmod{13} \)
- \( 10^{29} \equiv 6 \times 10 \equiv 60 \equiv 8 \pmod{13} \)
- \( 10^{30} \equiv 8 \times 10 \equiv 80 \equiv 2 \pmod{13} \)
- \( 10^{31} \equiv 2 \times 10 \equiv 20 \equiv 7 \pmod{13} \)
- \( 10^{32} \equiv 7 \times 10 \equiv 70 \equiv 5 \pmod{13} \)
- \( 10^{33} \equiv 5 \times 10 \equiv 50 \equiv 11 \pmod{13} \)
- \( 10^{34} \equiv 11 \times 10 \equiv 110 \equiv 6 \pmod{13} \)
- \( 10^{35} \equiv 6 \times 10 \equiv 60 \equiv 8 \pmod{13} \)
- \( 10^{36} \equiv 8 \times 10 \equiv 80 \equiv 2 \pmod{13} \)
- \( 10^{37} \equiv 2 \times 10 \equiv 20 \equiv 7 \pmod{13} \)
- \( 10^{38} \equiv 7 \times 10 \equiv 70 \equiv 5 \pmod{13} \)
- \( 10^{39} \equiv 5 \times 10 \equiv 50 \equiv 11 \pmod{13} \)
- \( 10^{40} \equiv 11 \times 10 \equiv 110 \equiv 6 \pmod{13} \)
- \( 10^{41} \equiv 6 \times 10 \equiv 60 \equiv 8 \pmod{13} \)
- \( 10^{42} \equiv 8 \times 10 \equiv 80 \equiv 2 \pmod{13} \)
- \( 10^{43} \equiv 2 \times 10 \equiv 20 \equiv 7 \pmod{13} \)
- \( 10^{44} \equiv 7 \times 10 \equiv 70 \equiv 5 \pmod{13} \)
- \( 10^{45} \equiv 5 \times 10 \equiv 50 \equiv 11 \pmod{13} \)
- \( 10^{46} \equiv 11 \times 10 \equiv 110 \equiv 6 \pmod{13} \)
- \( 10^{47} \equiv 6 \times 10 \equiv 60 \equiv 8 \pmod{13} \)
- \( 10^{48} \equiv 8 \times 10 \equiv 80 \equiv 2 \pmod{13} \)
- \( 10^{49} \equiv 2 \times 10 \equiv 20 \equiv 7 \pmod{13} \)
- \( 10^{50} \equiv 7 \times 10 \equiv 70 \equiv 5 \pmod{13} \)
- \( 10^{51} \equiv 5 \times 10 \equiv 50 \equiv 11 \pmod{13} \)
- \( 10^{52} \equiv 11 \times 10 \equiv 110 \equiv 6 \pmod{13} \)
- \( 10^{53} \equiv 6 \times 10 \equiv 60 \equiv 8 \pmod{13} \)
- \( 10^{54} \equiv 8 \times 10 \equiv 80 \equiv 2 \pmod{13} \)
- \( 10^{55} \equiv 2 \times 10 \equiv 20 \equiv 7 \pmod{13} \)
- \( 10^{56} \equiv 7 \times 10 \equiv 70 \equiv 5 \pmod{13} \)
- \( 10^{57} \equiv 5 \times 10 \equiv 50 \equiv 11 \pmod{13} \)
- \( 10^{58} \equiv 11 \times 10 \equiv 110 \equiv 6 \pmod{13} \)
- \( 10^{59} \equiv 6 \times 10 \equiv 60 \equiv 8 \pmod{13} \)
- \( 10^{60} \equiv 8 \times 10 \equiv 80 \equiv 2 \pmod{13} \)
- \( 10^{61} \equiv 2 \times 10 \equiv 20 \equiv 7 \pmod{13} \)
- \( 10^{62} \equiv 7 \times 10 \equiv 70 \equiv 5 \pmod{13} \)
- \( 10^{63} \equiv 5 \times 10 \equiv 50 \equiv 11 \pmod{13} \)
- \( 10^{64} \equiv 11 \times 10 \equiv 110 \equiv 6 \pmod{13} \)
- \( 10^{65} \equiv 6 \times 10 \equiv 60 \equiv 8 \pmod{13} \)
- \( 10^{66} \equiv 8 \times 10 \equiv 80 \equiv 2 \pmod{13} \)
- \( 10^{67} \equiv 2 \times 10 \equiv 20 \equiv 7 \pmod{13} \)
- \( 10^{68} \equiv 7 \times 10 \equiv 70 \equiv 5 \pmod{13} \)
- \( 10^{69} \equiv 5 \times 10 \equiv 50 \equiv 11 \pmod{13} \)
- \( 10^{70} \equiv 11 \times 10 \equiv 110 \equiv 6 \pmod{13} \)
- \( 10^{71} \equiv 6 \times 10 \equiv 60 \equiv 8 \pmod{13} \)
- \( 10^{72} \equiv 8 \times 10 \equiv 80 \equiv 2 \pmod{13} \)
- \( 10^{73} \equiv 2 \times 10 \equiv 20 \equiv 7 \pmod{13} \)
- \( 10^{74} \equiv 7 \times 10 \equiv 70 \equiv 5 \pmod{13} \)
- \( 10^{75} \equiv 5 \times 10 \equiv 50 \equiv 11 \pmod{13} \)
- \( 10^{76} \equiv 11 \times 10 \equiv 110 \equiv 6 \pmod{13} \)
- \( 10^{77} \equiv 6 \times 10 \equiv 60 \equiv 8 \pmod{13} \)
- \( 10^{78} \equiv 8 \times 10 \equiv 80 \equiv 2 \pmod{13} \)
- \( 10^{79} \equiv 2 \times 10 \equiv 20 \equiv 7 \pmod{13} \)
- \( 10^{80} \equiv 7 \times 10 \equiv 70 \equiv 5 \pmod{13} \)
- \( 10^{81} \equiv 5 \times 10 \equiv 50 \equiv 11 \pmod{13} \)
- \( 10^{82} \equiv 11 \times 10 \equiv 110 \equiv 6 \pmod{13} \)
- \( 10^{83} \equiv 6 \times 10 \equiv 60 \equiv 8 \pmod{13} \)
- \( 10^{84} \equiv 8 \times 10 \equiv 80 \equiv 2 \pmod{13} \)
- \( 10^{85} \equiv 2 \times 10 \equiv 20 \equiv 7 \pmod{13} \)
- \( 10^{86} \equiv 7 \times 10 \equiv 70 \equiv 5 \pmod{13} \)
- \( 10^{87} \equiv 5 \times 10 \equiv 50 \equiv 11 \pmod{13} \)
- \( 10^{88} \equiv 11 \times 10 \equiv 110 \equiv 6 \pmod{13} \)
- \( 10^{89} \equiv 6 \times 10 \equiv 60 \equiv 8 \pmod{13} \)
- \( 10^{90} \equiv 8 \times 10 \equiv 80 \equiv 2 \pmod{13} \)
- \( 10^{91} \equiv 2 \times 10 \equiv 20 \equiv 7 \pmod{13} \)
- \( 10^{92} \equiv 7 \times 10 \equiv 70 \equiv 5 \pmod{13} \)
- \( 10^{93} \equiv 5 \times 10 \equiv 50 \equiv 11 \pmod{13} \)
- \( 10^{94} \equiv 11 \times 10 \equiv 110 \equiv 6 \pmod{13} \)
- \( 10^{95} \equiv 6 \times 10 \equiv 60 \equiv 8 \pmod{13} \)
- \( 10^{96} \equiv 8 \times 10 \equiv 80 \equiv 2 \pmod{13} \)
- \( 10^{97} \equiv 2 \times 10 \equiv 20 \equiv 7 \pmod{13} \)
- \( 10^{98} \equiv 7 \times 10 \equiv 70 \equiv 5 \pmod{13} \)
- \( 10^{99} \equiv 5 \times 10 \equiv 50 \equiv 11 \pmod{13} \)
- \( 10^{100} \equiv 11 \times 10 \equiv 110 \equiv 6 \pmod{13} \)
- \( 10^{101} \equiv 6 \times 10 \equiv 60 \equiv 8 \pmod{13} \)
- \( 10^{102} \equiv 8 \times 10 \equiv 80 \equiv 2 \pmod{13} \)
- \( 10^{103} \equiv 2 \times 10 \equiv 20 \equiv 7 \pmod{13} \)
- \( 10^{104} \equiv 7 \times 10 \equiv 70 \equiv 5 \pmod{13} \)
- \( 10^{105} \equiv 5 \times 10 \equiv 50 \equiv 11 \pmod{13} \)
- \( 10^{106} \equiv 11 \times 10 \equiv 110 \equiv 6 \pmod{13} \)
- \( 10^{107} \equiv 6 \times 10 \equiv 60 \equiv 8 \pmod{13} \)
- \( 10^{108} \equiv 8 \times 10 \equiv 80 \equiv 2 \pmod{13} \)
- \( 10^{109} \equiv 2 \times 10 \equiv 20 \equiv 7 \pmod{13} \)
- \( 10^{110} \equiv 7 \times 10 \equiv 70 \equiv 5 \pmod{13} \)
- \( 10^{111} \equiv 5 \times 10 \equiv 50 \equiv 11 \pmod