Giải thích các bước giải:
Bài 1:
a.Ta có:
$5n+7\quad\vdots\quad n+2$
$\to (5n+10)-3\quad\vdots\quad n+2$
$\to 5(n+2)-3\quad\vdots\quad n+2$
$\to 3\quad\vdots\quad n+2$
$\to n+2\in U(3)$
Mà $n+2\ge 0+2=2\to n+2=3\to n=1$
b.Để $3-n\quad\vdots\quad n+1$
$\to 4-n-1\quad\vdots\quad n+1$
$\to 4-(n+1)\quad\vdots\quad n+1$
$\to 4\quad\vdots\quad n+1$
$\to n+1\in U(4)=\{1,2,4\}$ vì $n\in N\to n\ge 0\to n+1\ge 1$
$\to n\in\{0,1,3\}$
Bài 2:
a.Ta có:
$5,5^2,5^3,...,5^{100}\quad\vdots\quad 5$
$\to A=5+5^2+5^3+...+5^{100}\quad\vdots\quad 5$
b.Ta có:
$A=5+5^2+5^3+...+5^{100}$
$\to A=5(1+5+5^2+...+5^{99})$
Do $1+5+5^2+...+5^{99}\quad\not\vdots\quad 5$
$\to A\quad\not\quad\vdots\quad 25$
Mà $A\quad\quad 5$
$\to A$ không là số chính phương
c.Ta có:
$A=5+5^2+5^3+...+5^{100}$
$\to 5A=5^2+5^3+5^4+...+5^{101}$
$\to 5A-A=5^{101}-5$
$\to 4A=5^{101}-5$
$\to 4A+5=5^{101}$
$\to 25^a.5=5^{101}$
$\to (5^2)^a.5=5^{101}$
$\to 5^{2a}.5=5^{101}$
$\to 5^{2a+1}=5^{101}$
$\to 2a+1=101$
$\to 2 a=100$
$\to a=50$
