Finding the value of abc

Question

The ratio of the six-digit numbers $abcabc$ and $ababab$ is 55:54. Find the values of the digits $a$, $b$ and $c$.

Answer 1

$abcabc = abc\times 1001 = abc\times 7\times11\times 13$

$ababab = ab\times 10101 = ab\times 3\times 7\times 13\times 37$

$\frac{abcabc}{ababab} = \frac{abc}{ab}\frac{11}{3\times 37} = \frac{55}{54} \implies \frac{abc}{ab} = \frac{185}{18}$

... which is conveniently the form we need (we don't need to multiply both sides by some constant to make the digits match up). So $abcabc = 185185$ and $ababab = 181818$.

Answer 2

The easy way -- from the problem statement

$$5405400A+540540B+54054c=555550A+55555B $$ This gives $$ 54054C = 150150A + 15015B $$ The RHS is divisible by $5$ so $C$ is divisible by $5$ so $C=5$.

If $C=5$ the LHS is $270270 < 150150\cdot 2$ so $A=1$.

Finally $15015B = 270270-150150 = 120120$ so $B=8$.