isi-entrance 2015 QB4

isi-entrance · India · UGB Number Theory Divisibility and Divisor Analysis

Let $a _ { n } = 1 \ldots 1$ with $3 ^ { n }$ digits. Prove that $a _ { n }$ is divisible by $3 a _ { n - 1 }$.

Observe that $a _ { n } = a _ { n - 1 } \left( 1 + t + t ^ { 2 } \right)$ where $t = 10 ^ { 3 ^ { n } }$.

Let $a _ { n } = 1 \ldots 1$ with $3 ^ { n }$ digits. Prove that $a _ { n }$ is divisible by $3 a _ { n - 1 }$.