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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?

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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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  • $\begingroup$ So from what i understood we have to first show that 20 is a unit of modulo 123 and then find the inverse of 20 which is 80 and by $80^{-1}.20=1$ ($\mod 123$) we can say $80^{-1}.20x=1.x=k.11$ ($\mod 123$). But what does x mean in our answer is it the number of schools?and how do we find how many talk were given? $\endgroup$ – user102867 Dec 8 '13 at 19:27
  • $\begingroup$ The inverse of $20$ is $80$, not $80^{-1}$. So multiplying both sides of the congruence by $80$ gives us $80\cdot 20x = 1x = x \equiv 80\cdot 11 \pmod{123}$. $x$ is the number of schools here. $\endgroup$ – amWhy Dec 8 '13 at 19:54
  • $\begingroup$ Thanks I have found x as 19 so what about the number of talks? Is it 4 because each group has 20 students and there are 19 schools so total number of students are 380 and theatre has a space of 123 and we know that the last lecture has 11 students so it should be $3*123+11=20*19$ so the formula is $20x-123l=11$. Am i right and is this the way to find the number of talks? $\endgroup$ – user102867 Dec 8 '13 at 21:20
  • $\begingroup$ Yes, indeed you are correct! Nice work. ;-) $\endgroup$ – amWhy Dec 8 '13 at 21:22
  • $\begingroup$ @amWhy: Needs another UV +1 $\endgroup$ – Amzoti Dec 9 '13 at 1:11

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