(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Basically:

Olivia pays 234 kr for a train ticket, and John pays 264 kr for the same kind of ticket. After half a year, Olivia has paid 54 kr more than John. How many tickets did they buy each?

By the way, kr is the currency in Norway.

2. Relevant equations

Uhm, I guess:

ax + ny = b <=> ax is congruent with b (mod n)

and all the usual equations for congruences.

3. The attempt at a solution

This is what I've done (I agree with the solution [which I have] quite far, but there's one bit I don't understand):

234x = 264y + 54

234x is congruent with 54 (mod 264) => gcd(234, 264) = 6 => 6|54

39x is congruent with 9 (mod 44)

(9 + 44n)/39 has to be a natural number. n = 6

39x is congruent with 9 + 44 * 6 (mod 44)

39x is congruent with 273 (mod 44)

x is congruent with 7 (mod 44)

x = 7 + 44n

Okay, the solution agrees with me this far. Then I got blank, as this is supposed to have 6 incongruences modulo 264, so I checked it up in the solution. It says: "If Olivia takes the train every weekend, x_(max) = 24 => x = 7 + 44 * 0 = 7" and then finds y = 6.

But I didn't understand that about xmax being 24, and why it apparently doesn't matter at all that it's 24. Why does it tell me to continue like that? I understand everything that's done, except from that tiny part about deciding that n=0 is the only possible solution for x = 7 + 44n.

Thanks :)

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# Homework Help: Linear congruence

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