Question

1.Aratati ca 100| a ,unde a=7^n+7^n+1+7^n+2+7^n+3,oricare ar fi n∈N.
2.Aratati ca 19| a ,unde a=3^n*5^n+2+3^n+2*5^n-3^n+1*5^n+1,oricare ar fi n∈N.

Answer 1

1. a=7∧n+7∧n+1+7∧n+2+7∧n+3
a=7∧n(1+7+7²+7³)
a=7∧n(1+7+49+343)
a=7∧n·400
Dar deoarece 400=4·100⇒100 divide 400 atunci evident 100 divide pe a.
2. a=3∧n·5∧n+2+3∧n+2·5∧n-3∧n+1·5∧n+1
a=(3∧n·5∧n)(5²+3²-3·5)
a=(3∧n·5∧n)(25+9-15)
a=(3∧n·5∧n)·19
Dar deoarece 19=1·19⇒19 divide 19 atunci evident 19 divide pe a.