Question

Stabiliţi care dintre numerele x şi y au mai mulți divizori:
a) x=450; y=324;
b)x=150; y=120;
c)x=72; y=125;
d) x=350; y=400.

Answer 1

Răspuns: Ai mai jos rezolvarea pentru fiecare punct in parte

$x = 450=2\cdot 3^{2} \cdot 5^{2}\Rightarrow \boldsymbol{\tau}_{D_{450}}=(1+1)\cdot(2+1)\cdot(2+1)\Rightarrow \boxed{\boldsymbol{\tau}_{D_{450}}=18}$

$y = 324=2^{2}\cdot 3^{4} \Rightarrow \boldsymbol{\tau}_{D_{324}}=(2+1)\cdot(4+1)\Rightarrow \boxed{\boldsymbol{\tau}_{D_{324}}=15}$

$\red{ \boxed{~\boldsymbol{\tau}_{D_{450}}~>~\boldsymbol{\tau}_{D_{324}}~}}$

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$x = 150=2\cdot 3 \cdot 5^{2}\Rightarrow \boldsymbol{\tau}_{D_{150}}=(1+1)\cdot(1+1)\cdot(2+1)\Rightarrow \boxed{\boldsymbol{\tau}_{D_{150}}=12}$

$y = 120=2^{3}\cdot 3\cdot 5\Rightarrow \boldsymbol{\tau}_{D_{120}}=(3+1)\cdot(1+1)\cdot(1+1)\Rightarrow \boxed{\boldsymbol{\tau}_{D_{120}}=16}$

$\purple{ \boxed{~\boldsymbol{\tau}_{D_{150}}~<~\boldsymbol{\tau}_{D_{120}}~}}$

____________________

$x = 72=2^{3}\cdot 3^{2} \Rightarrow \boldsymbol{\tau}_{D_{72}}=(3+1)\cdot(2+1)\Rightarrow \boxed{\boldsymbol{\tau}_{D_{72}}=12}$

$y = 125=5^{3}\Rightarrow \boldsymbol{\tau}_{D_{125}}=(3+1)\Rightarrow \boxed{\boldsymbol{\tau}_{D_{125}}=4}$

$\blue{ \boxed{~\boldsymbol{\tau}_{D_{72}}~>~\boldsymbol{\tau}_{D_{125}}~}}$

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$x = 350=2\cdot 5^{2} \cdot 7\Rightarrow \boldsymbol{\tau}_{D_{350}}=(1+1)\cdot(2+1)\cdot(1+1)\Rightarrow \boxed{\boldsymbol{\tau}_{D_{350}}=12}$

$y = 400=2^{4}\cdot 5^{2}\Rightarrow \boldsymbol{\tau}_{D_{400}}=(4+1)\cdot (2+1)\Rightarrow \boxed{\boldsymbol{\tau}_{D_{400}}=15}$

$\pink{ \boxed{~\boldsymbol{\tau}_{D_{350}}~<~\boldsymbol{\tau}_{D_{400}}~}}$