- #1
Norway
- 50
- 3
Homework Statement
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.
Homework Equations
Uhm, I guess:
ax + ny = b <=> ax is congruent with b (mod n)
and all the usual equations for congruences.
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 :)