Question

Care din nr 10125 si 3136 are mai multi divizori intregi

Answer 1

 

$\displaystyle\\\text{Un numar natural are de 2 ori mai multi divizori intregi decat naturali.}\\\text{Descompunem numerele in factori primi.}$

.

$\displaystyle\\10125~\vdots~125\\\\\frac{10125}{125}=81\\\\ \implies~10125=81\times125\\\\\boxed{\bf10125=3^4\times5^3}\\n=(4+1)(3+1)=5\times4=\boxed{\bf20~\text{\bf de divizori naturali}}\\\implies~2n=2\times20=\boxed{\bf40~\text{\bf de divizori intregi}}\\\\\\3136~\text{este patrat perfect.}\\3136=56^2\\3136=(8\times7)^2\\3136=8^2\times7^2\\\\\boxed{\bf3136=2^6\times7^2}\\\\n=(6+1)(2+1)=7\times3=\boxed{\bf21~de~divizori~naturali}\\2n=2\times21=\boxed{\bf42~de~divizori~intregi}$

.

$\displaystyle\bf\\42>40\\\\\implies~\boxed{\bf3136~~\text{\bf are mai multi divizori intregi decat }~10125}$