Competition problem — Number theory

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Number theory Difficulty 6.2 National olympiad Prove it

20. Let $k \geqslant 1$ . Prove:
(i) If $2^{k} \leqslant n<2^{k+1}$ , and $1 \leqslant a \leqslant n, a \neq 2^{k}$ , then $2^{k} \nmid a$ ;
(ii) If $3^{k} \leqslant 2 n-1<3^{k+1}, 1 \leqslant l \leqslant n, 2 l-1 \neq 3^{k}$ , then $3^{k} \nmid 2 l-1$ .

Solution

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