Kỹ thuật đồng dư trong APMOPS 2004


Question:

A three digit number \overline{5ab} is written 99 times as A=5ab5ab5ab…5ab.
The resultant number is a multiple of 91.
What is the three digit number?

Solution:

It is easy to prove that A=\overline{5ab}(10^{3.98}+10^{3.97}+...+10^3+1). Observe that 10^3=1000\equiv -1 (\emph{ mod 91}).

Thus 10^{3.98}+10^{3.97}+...+10^3+1\equiv 1(\emph{ mod 91}) \Rightarrow 91\mid \overline{5ab}.

Now we can find that \overline{5ab}=546.



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