\[{\text{Find number }}a > 1,{\text{ if numbers 1010 and 1069 produce equal remainders when divided by }}a.\]
\[\begin{array}{l}
1010 = m \cdot a + r \hfill \\
1069 = n \cdot a + r \hfill \\
{\text{so }}1069 - 1010 = \left( {m - n} \right) \cdot a = 59{\text{ is divisible by }}a. \hfill \\
{\text{But 59 is a prime number}} \Rightarrow a = 59. \hfill \\
\end{array} \]
59
