Question

Determinați cel mai mare număr natural n astfel încât 1 ×3 × 5 ... × 100 divizibil cu 3 la puterea n

Answer 1

$1\times 3\times 5\times7\times 9\times 11\times 13\times 15\times 17\times 19\times 21\times...\times 97\times 99\times 100 = \\ \\ = .(3\times1)\times ....\times (3\times 3)\times...\times (3\times 5)\times...\times (3\times 7)\times...\times (3\times 33)\times\\ \times 100\\ \\ De la 1,3,5,...,pana la 33 \Rightarrow 1,3,5,..,(2\cdot 17-1) \rightarrow avem 17 termeni. \\ Scoatem:$

$3^{17}\times 1\times 3\times 5\times 7\times ...\times 33 =\\ = 3^{17}\times...(3\times1)\times...(3\times 3)\times...\times (3\times 11)\\ \\ De la 1,3,5,,,11 avem 6 termeni. \\ Scoatem: \\ \\ 3^{17}\times 3^{6}\times 1\times 3\times 5\times...\times 11 = \\ = 3^{17}\times 3^{6}..\times(3\times 1)\times...\times(3\times 3)\times... \\ \\ De la 1,3, avem 2 termeni\\ Scoatem: \\ \\ 3^{17}\times 3^{6}\times 3^{2}\times 1\times 3 = 3^{17}\times 3^{6}\times 3^{2}\times3^1=3^{17+6+2+1} =3^{26}$

=> n = 26.