Thu May 25, 2006 3:04 am
Here's a mathematical solution:
Let's say the total number of apples produced by each farmer averages n. (Trap: Note here that our answer is going to be 15*n, not n.)
Because there can be only a whole number of apples produced by each orchard, we know that n must be divisible by 1, 2, 3, 4, 5, 6...8...10...12...and 15. (We'll deal with the farmers who exchange apples later.) In other words:
n is congruent to 0 mod 1, 0 mod 2, 0 mod 3, 0 mod 4, 0 mod 5, 0 mod 6, 0 mod 8, 0 mod 10, 0 mod 12, and 0 mod 15.
The smallest number that satisfies the above conditions is LCM(1, 2, 3, 4, 5, 6, 8, 10, 12, 15) = LCM(8, 10, 12, 15) = 30.
From the first exchange of apples, we know that n is one short of being divisible by 11, or:
n is congruent to -1 mod 11, which is also congruent to 10 mod 11 by the definition of modulus.
Continuing on in a similar manner, we know that:
n is congruent to 1 mod 7 (statement 1)
n is congruent to -6 mod 14 = 8 mod 14 (statement 2)
n is congruent to 3 mod 9
n is congruent to 3 mod 13
Timesaver: Notice that statement 2 implies statement 1, so satsifying statement 2 will satisfy statement 1.
I arranged the above facts into a table like this:
Find mod(30, 9) = 3. This happens to be what we're looking for.
Find mod(30, 11) = 8. This needs to become a 10.
To retain our results so far, we need to find the smallest number that is 0 mod 30 and 0 mod 9, but NOT 0 mod 11. Then we add that to our answer so far, 30, which "dials" the mod 11 column to a different number. That number is LCM(30, 9) = 90, which is congruent to 2 mod 11.
We get 10 in the "mod 11" column, so we're done. (FYI, if we didn't get the right number, we would have added 90 again until we did.)
We then calculate:
mod(120, 13) = 3
mod(120, 14) = 8
These happen to already be correct, so we've found an n that satisfies all the conditions.
15 * n = 15 * 120 = 1800
Multiple answers: Note that if we take add LCM(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15) = 360360, we get yet another very large n that works. However, this would mean that 360480 apples grew on the first farmer's single tree! This doesn't make sense.
That was fun.
Let's say the total number of apples produced by each farmer averages n. (Trap: Note here that our answer is going to be 15*n, not n.)
Because there can be only a whole number of apples produced by each orchard, we know that n must be divisible by 1, 2, 3, 4, 5, 6...8...10...12...and 15. (We'll deal with the farmers who exchange apples later.) In other words:
n is congruent to 0 mod 1, 0 mod 2, 0 mod 3, 0 mod 4, 0 mod 5, 0 mod 6, 0 mod 8, 0 mod 10, 0 mod 12, and 0 mod 15.
The smallest number that satisfies the above conditions is LCM(1, 2, 3, 4, 5, 6, 8, 10, 12, 15) = LCM(8, 10, 12, 15) = 30.
From the first exchange of apples, we know that n is one short of being divisible by 11, or:
n is congruent to -1 mod 11, which is also congruent to 10 mod 11 by the definition of modulus.
Continuing on in a similar manner, we know that:
n is congruent to 1 mod 7 (statement 1)
n is congruent to -6 mod 14 = 8 mod 14 (statement 2)
n is congruent to 3 mod 9
n is congruent to 3 mod 13
Timesaver: Notice that statement 2 implies statement 1, so satsifying statement 2 will satisfy statement 1.
I arranged the above facts into a table like this:
- Code:
Goal: 0 3 10 3 8
n mod 30 mod 9 mod 11 mod 13 mod 14
30 0
Find mod(30, 9) = 3. This happens to be what we're looking for.
Find mod(30, 11) = 8. This needs to become a 10.
To retain our results so far, we need to find the smallest number that is 0 mod 30 and 0 mod 9, but NOT 0 mod 11. Then we add that to our answer so far, 30, which "dials" the mod 11 column to a different number. That number is LCM(30, 9) = 90, which is congruent to 2 mod 11.
- Code:
Goal: 0 3 10 3 8
n mod 30 mod 9 mod 11 mod 13 mod 14
30 0 3 *8*
(+90) (+0) (+0) (+2)
120 0 3 10
We get 10 in the "mod 11" column, so we're done. (FYI, if we didn't get the right number, we would have added 90 again until we did.)
We then calculate:
mod(120, 13) = 3
mod(120, 14) = 8
These happen to already be correct, so we've found an n that satisfies all the conditions.
15 * n = 15 * 120 = 1800
Multiple answers: Note that if we take add LCM(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15) = 360360, we get yet another very large n that works. However, this would mean that 360480 apples grew on the first farmer's single tree! This doesn't make sense.
That was fun.