Find all three-digit numbers $overline{abc}$ such that 6003 digit number $overline{abcabcabc.....abc}$ is divisible by 91?

Find all three-digit numbers $\overline{abc}$ such that $6003$ digit number $\overline{abcabcabc......abc}$ is divisible by 91?Here $\overline{abc}$ occurs $2001$ times.I know the divisibility rule for 91 which states to subtract $9$ times the last digit from the rest and, for large numbers,to form the alternating sum of blocks of three numbers from right to left. However, I am not able to see how I could apply this rule to determine the numbers $\overline{abc}$. How can I solve this?

Answer

The given number can be written as follows,

$abc(1+10^3+10^6+\cdots+10^{6000})$

Now, $91|1001=1+10^3$ . The sum $S=1+10^3+10^6+\cdots+10^{6000}$ has $2001$ terms, therefore, $91$ and $(1+10^3)+10^6(1+10^3)+\cdots+10^{1999}(1+10^3)+10^{6000}$ are relatively prime $\implies$ $abc$ is a multiple of 91.

Therefore, the required numbers are $91\times n$ , where $n={2,3,4,5,6,7,8,9}$ i.e., the required numbers are :

$182,273,364,455,546,637,728,819$ and $910$