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1+3+a+b+4+5+6\\ =19+a+b\\\quad if\;\;a=8+b\;\;then\\ \\\quad =19+b+8+b\\\quad  =27+2b\\ \quad b=0\;\;or\;\;9\\ b=9 \;\;doesn't \;\;work\\ so\;\;   b=0,\;\;a=8 \; \text{is a possible solution}\\ ~\\

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\(\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|} \hline \mathbf{a+b}\pmod{9}= 8 \ ? \\ b \rightarrow & \mathbf{0} & \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9}\\ a \downarrow \\ \hline \mathbf{0} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & \color{red}8 & 0 \\ \hline \mathbf{1} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & \color{red}8 & 0 & 1 \\ \hline \mathbf{2} & 2 & 3 & 4 & 5 & 6 & 7 & \color{red}8 & 0 & 1 & 2 \\ \hline \mathbf{3} & 3 & 4 & 5 & 6 & 7 & \color{red}8 & 0 & 1 & 2 & 3 \\ \hline \mathbf{4} & 4 & 5 & 6 & 7 & \color{red}8 & 0 & 1 & 2 & 3 & 4 \\ \hline \mathbf{5} & 5 & 6 & 7 & \color{red}8 & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \mathbf{6} & 6 & 7 & \color{red}8 & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \mathbf{7} & 7 & \color{red}8 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \mathbf{8} & \color{red}8 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7& \color{red}8 \\ \hline \mathbf{9} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7& \color{red}8 & 0 \\ \hline \end{array}\)

\(\mathbf{a+b}\pmod{9}= 8 : \quad (a,b)=\{(0,8),(1,7),(2,6),(3,5),(4,4),(5,3),(6,2),{\color{red}(8,0)},(8,9),(9,8)\} \\ \mathbf{a-b}\pmod{11}= 8 :\quad (a,b)=\{(0,3),(1,4),(2,5),(3,6),(4,7),(5,8),(6,9),{\color{red}(8,0)},(9,1)\}\)

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thank you! yeah I forgot about adding a-b=8 and a+b=8 together. thank you so much for the work you put into the clear solution! :)

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\(\begin{array}{|rcll|} \hline 1+3+a+b+4+5+6 &\equiv& 0 \pmod{9} \\ a+b+19 &\equiv& 0 \pmod{9} \quad | \quad 19 \equiv 1 \pmod{9} \\ a+b+1 &\equiv& 0 \pmod{9} \quad | \quad - 1 \\ a+b &\equiv& -1 \pmod{9} \\ a+b &\equiv& -1+9 \pmod{9} \\ \mathbf{a+b} &\equiv& \mathbf{ 8 \pmod{9} } \\ \hline \end{array}\)

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\(\begin{array}{|lr|} \hline & \mathbf{13ab45c} = \mathbf{1380456} \\ \hline \text{check}: \\ & 1380456 : 792 = 1743 \\ \hline \end{array}\)

\(\begin{array}{|c|r|c|} \hline c & 45c & 45c \pmod{8} \equiv 0\ ?\\ \hline 2 & 452 & \text{no} \\ \hline 4 & 454 & \text{no} \\ \hline \mathbf{\color{red}6} & 456 & \text{yes} \\ \hline 8 & 458 & \text{no} \\ \hline \end{array} \), so \(\mathbf{c=6}\)

\(\begin{array}{|rcll|} \hline 1-3+a-b+4-5+6 &\equiv& 0 \pmod{11} \\ a-b+3 &\equiv& 0 \pmod{11} \quad | \quad - 3 \\ a-b &\equiv& -3 \pmod{11} \\ a-b &\equiv& -3+11 \pmod{11} \\ \mathbf{a-b} &\equiv& \mathbf{ 8 \pmod{11} } \\ \hline \end{array}\)

\(-9\le a-b\le9\\ -6\le 3+a-b \le12\\ which\;\; means\\ 3+a-b=11\;\;or\;\;0\\ a-b=8\;\;or\;\;-3\\ a=8+b\;\;or \;\; a=-3+b\)

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for it to be divisible by 11, 1-3+a-b+4-5+6= multiple of 11, so 3+a-b= multiple of 11, which means that a-b=8 or a-b = -3.

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\(\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|} \hline \mathbf{a-b}\pmod{11}= 8 \ ? \\ b \rightarrow & \mathbf{0} & \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9}\\ a \downarrow \\ \hline \mathbf{0} & 0 & 10 & 9 & \color{red}8 & 7 & 6 & 5 & 4 & 3 & 2 \\ \hline \mathbf{1} & 1 & 0 & 10 & 9 & \color{red}8 & 7 & 6 & 5 & 4 & 3 \\ \hline \mathbf{2} & 2 & 1 & 0 & 10 & 9 & \color{red}8 & 7 & 6 & 5 & 4 \\ \hline \mathbf{3} & 3 & 2 & 1 & 0 & 10 & 9 & \color{red}8 & 7 & 6 & 5 \\ \hline \mathbf{4} & 4 & 3 & 2 & 1 & 0 & 10 & 9 & \color{red}8 & 7 & 6 \\ \hline \mathbf{5} & 5 & 4 & 3 & 2 & 1 & 0 & 10 & 9 & \color{red}8 & 7 \\ \hline \mathbf{6} & 6 & 5 & 4 & 3 & 2 & 1 & 0 & 10 & 9 & \color{red}8 \\ \hline \mathbf{7} & 7 & 6 & 5 & 4 & 3 & 2 & 1 & 0 & 10 & 9 \\ \hline \mathbf{8} & \color{red}8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 & 0 & 10 \\ \hline \mathbf{9} & 9 & \color{red}8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 & 0 \\ \hline \end{array}\)

\(1+3+a+b+4+5+6\\ =19+a+b\\\quad if\;\;a=8+b\;\;then\\ \\\quad =19+b+8+b\\\quad =27+2b\\ \quad b=0\;\;or\;\;9\\ b=9 \;\;doesn't \;\;work\\ so\;\; b=0,\;\;a=8 \; \text{is a possible solution}\\ ~\\ 1+3+a+b+4+5+6\\ =19+a+b\\\quad if\;\;a=-3+b\;\;then\\ \\\quad =19+b-3+b\\\quad =16+2b\\ \quad b=1,\;\;a=-2 \;\text{which is invalid}\)

1+3+a+b+4+5+6\\ =19+a+b\\\quad if\;\;a=-3+b\;\;then\\ \\\quad =19+b-3+b\\\quad  =16+2b\\  \quad b=1,\;\;a=-2 \;\text{which is invalid}

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