General Tso was leading a phalanx of chickens and knew there were between 1000 and 2000 in all. He desired a more accurate estimate, and so had a lieutenant line them up in columns of 11, 13, and 17 and use the Chinese Remainder Theorem to obtain a good tally for the general. The general was very happy with the resulting numbers until he found out that before the first two assemblies, Chicken Little was still sleeping. If Chicken Little made the third assembly (and was the only chicken still in the coop), how wrong were General Tso’s “more accurate” numbers?
The wording of the question is a little difficult, but the way I am interpreting this question is that the chickens are lined up in 3 columns/assemblies. The first column/assembly has 11 chickens, the second column/assembly has 13 chickens, and the third column/assembly has 17 chickens. One paricular chicken does not show up in the lineup for 11 or 13 but showed up for the assembly of 17.
So this is what I think the system of congruences will look like:
$x$ $\equiv$ 1 mod 11
$x$ $\equiv$ 1 mod 13
$x$ $\equiv$ 0 mod 17
Then, we solve for the system of congruences.
$x = 17a + 0$
$17a + 0 \equiv 1 \ (mod \ 13)$
$4a \equiv 1 \ (mod \ 13)$
$40 a \equiv 10 \ (mod \ 13) $
$a \equiv 13b + 10 $
$x = 17(13b + 10) = 221b + 10$
$x \equiv 1 \ (mod \ 11)$
$221b = 10 = 1 \ (mod \ 11)$
$b + 10 = 1 \ (mod \ 11)$
$b = -9 \ (mod \ 11)$
$b = 2 \ (mod \ 11)$
$b = 11c + 2$
$221(11c + 2) + 10 = 2431c + 452$.
Did I do something wrong in my calculations, because my remainder is not between 1000 and 2000. Or is my entire setup incorrect?