All-Russian Olympiad Problem 5

@Adarsh Kumar – Have you understood Andrei's solution upto step 4 ? The factorisation of $6 \times 700 = 6 \times 7 \times 4 \times 25$ and GCD( $6 \times 7 \times 4, 7q_{1}+6q_{2}$ ) = 1 made me take $25|7q_{1}+6q_{2}$ as a suitable constraint. We can see that minimum integral values of $q_{1}$ and $q_{2}$ such that $25|7q_{1}+6q_{2}$ is attained at least integral solution ( $q_{1}, q_{2}$ ) for the diophantine equation $7q_{1}+6q_{2}=25$ , and I think you very well know how to solve this ( A standard diophantine equation of form $ax+by=c$ ).We come to know from Euclid's algorithm that the least solution set obeying the equation is $(q_{1}, q_{2})$ = ( 1,3 ). [ If you do not know how to solve diophantine equations, look for it in Number theory Wikis in Brilliant]. Are you clear now ?