Why is くなかった wrong and ではありません preferred in this sentence?

Question

I was taking a n5 grammar test and I came across the question:

あの人{ひと} は有名{ゆうめい}　＿＿＿＿

The answer I chose was くなかった. However, the quiz said the correct answer was ではありません. Why is that?

Answer 1

有名 is a na-adjective and thus is followed by だ・で・な・に (or by じゃ < では)

あの人は有名だ
lit. that person is famous

あの人は有名な人だ
lit. that person is a famous person

あの人はきっと有名になる
lit. that person is surely going to become famous

You can think of ではありません or じゃありません as the negative of です (which in turn is the polite form of だ whose negative would be ではない or じゃない). Hence

あの人は有名ではありません
lit. that person is not famous

The point of the question seems to be to test your ability in identifying i- and na-adjectives correctly as well as inflecting these adjectives. One cheap way of recognizing i-adjectives is by their dictionary form, which always ends in an okurigana い (like 速い、寒い、安い、…). Here 有名 does not inflect like an i-adjective. Of course, i-adjectives would inflect like

あの人は速くなかった
lit. that person was not fast

where くなかった is the ending you chose. Here 速い hayai means "fast" (note the い as okurigana).