Problems

Solution: The first digit is \(2\) or \(5\), all the other digits can be any value from \(0,2,5,7\). Therefore we have \begin{align*} S &= 2000 \cdot 4^3+5000 \cdot 4^3 + (200+500+700) \cdot 2 \cdot 4^2 + (20+50+70) \cdot 2 \cdot 4^2 + (2+5+7) \cdot 2 \cdot 4^2 \\ &= 7 \cdot 4^3 \cdot 2^3 \cdot 5^3 + 14 \cdot 2 \cdot 4^2 \cdot 111 \\ &= 2^{9} \cdot5^3 \cdot 7 + 2^{6} \cdot 3 \cdot 7 \cdot 37 \\ &= 2^6 \cdot 7 \cdot (1000+111) \\ &= 2^6 \cdot 7 \cdot 11 \cdot 101 \end{align*} Alternatively, consider adding the first and last terms, and second and second and last terms, etc we obtain \(7777\). There are \(2 \cdot 4^3\) terms so \(7777 \cdot 4^3 = 2^6 \cdot 7 \cdot 11 \cdot 101\)