Local secondary school send groups of twenty students to the university for a summer school. During this event, every student is scheduled to have a careers talk in the theatre, which has a seated capacity of one hundred and twenty-three. If the final talk must be delivered only to eleven students, but every preceding talk was filled to capacity, then how many schools sent students to summer school? How many talks were given in total?
I just don't understand what number goes where. What i think is we have 20x=11 mod 123 or 20x=1 mod 123 where x is the inverse of 20 mod 123 which is 80 mod 123 but is 11 the remainder and how do we solve this question? Can someone explain please?
Answer
The correct congruence equation here is $$20x\equiv 11 \pmod{123}$$
To solve for $x$, multiply both sides of the congruence equation by the multiplicative inverse $k = 20^{-1}$ of $20$, modulo $123$. Then $$k\cdot 20 x \equiv x\equiv k \cdot 11 \pmod {123}$$
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