This Olympiad questions states:

[Show that every term of the sequence 10001, 100010001, 1000100010001, ... is composite.]

Well, I figured out this sequence is generalised as a_{n}= 10001^{n}for all n $\displaystyle >=$ 1.

After actually doing this division in Excel(I know... I cheated), I found out the prime factorisation of 10001 = 73*137, hence proving the claim.

If I had no access to Excel, how would you have come up with this factorisation. Any results we can make use of to get the prime factorisation?

Thanks