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2.

(1)15個　(2)16個　(3)17個　(4)18個　(5)19個

What is the largest 2-digit prime factor of the integer $$\left(\matrix{200\cr 100}\right)$$？
(1983AIME，https://artofproblemsolving.com/ ... _Problems/Problem_8)

B.

D.

G.

I.

$$\matrix{3&5&7&9&11&13&15&17&19&\ldots\cr 4&7&10&13&16&19&22&25&28&\ldots\cr 5&9&13&17&21&25&29&33&37&\ldots\cr 6&11&16&21&26&31&36&41&46&\ldots\cr 7&13&19&25&31&37&43&49&55&\ldots\cr 8&15&22&29&36&43&50&57&64&\ldots\cr 9&17&25&33&41&49&57&65&73&\ldots\cr \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&}$$
[解答]
$$\matrix{a_0&&a_1&&a_2&&a_3&&a_4&&a_5&&a_6&&a_7\cr 1&&3&&7&&13&&21&&31&&43&&57\cr &2&&4&&6&&8&&10&&12&&14&\cr &&2&&2&&2&&2&&2&&2&&}$$
$$a_n=1\times C_0^n+2\times C_1^n+2\times C_2^n=n^2+n+1$$
$$a_{20}=421$$

J.

https://math.pro/db/attachment.php?aid=6086&k=dfa7eb4f0db6f64b2b485b4f118cc721&t=1695826065

https://math.pro/db/attachment.php?aid=6088&k=a169ab15aaf5560f4077b23eede34fce&t=1695826065

(自己看錯)

$$(7242409)^{10}$$除以$$101\times 102$$的餘數為

112.7.7補充

https://math.pro/db/attachment.php?aid=6099&k=6e8cb1a1869302315fa2d47183979d40&t=1695826065

https://math.pro/db/attachment.php?aid=6100&k=11c635583db1d360de34a9716aa8f61c&t=1695826065

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C.

[解答]

n^3 + 2n^2 - 200n
= n[n(n + 2) - 200]

1 <= n <= 13，a_n < 0
n >= 14，a_n > 0

H.

[解答]

112.7,7補充