Question

Cho A =350.(15²⁰⁰⁷ +15 ²⁰⁰⁶ +...........+15² +16) + 25

a, Rút gọn A

b, Chứng minh rằng A chia hết cho 5²⁰¹⁰

Answer 1

a)\(A-25=350.\left(15^{2007}+15^{2006}+...+15+1\right)\)

\(\frac{A-25}{350}=15^{2007}+15^{2006}+...+15+1\)

\(\frac{\left(A-25\right).15}{350}=15^{2008}+15^{2007}+...+15^2+15\)

\(\Rightarrow\frac{15.\left(A-25\right)}{350}-\frac{A-25}{350}=15^{2008}-1\)

\(\frac{15A-25.15-A+25}{350}=\frac{14A-25.14}{350}=15^{2008}-1\)

\(\frac{14\left(A-25\right)}{350}=15^{2008}-1\)

\(A-25=\frac{350\left(15^{2008}-1\right)}{14}=25.\left(15^{2008}-1\right)\)

\(\Rightarrow A=25.15^{2008}\)

b)15 chia hết cho 5 suy ra 15²⁰⁰⁸ chia hết cho 5²⁰⁰⁸

suy ra 25.15²⁰⁰⁸ chia hết cho 25.5²⁰⁰⁸=5²⁰¹⁰

Answer 2

a)\(A=25.15^{2008}\)

b)A=25.15²⁰⁰⁸ chia hết cho 25.5²⁰⁰⁸=5²⁰¹⁰ ,suy ra điều phải chứng minh