cmi-entrance 2012 QB7

cmi-entrance · India · ugmath 10 marks Number Theory Congruence Reasoning and Parity Arguments
A sequence of integers $c _ { n }$ starts with $c _ { 0 } = 0$ and satisfies $c _ { n + 2 } = a c _ { n + 1 } + b c _ { n }$ for $n \geq 0$, where $a$ and $b$ are integers. For any positive integer $k$ with $\operatorname { gcd } ( k , b ) = 1$, show that $c _ { n }$ is divisible by $k$ for infinitely many $n$. 
A sequence of integers $c _ { n }$ starts with $c _ { 0 } = 0$ and satisfies $c _ { n + 2 } = a c _ { n + 1 } + b c _ { n }$ for $n \geq 0$, where $a$ and $b$ are integers. For any positive integer $k$ with $\operatorname { gcd } ( k , b ) = 1$, show that $c _ { n }$ is divisible by $k$ for infinitely many $n$.
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QA1 QA2 QA3 QA4 QA5 QB1 QB2 QB3 QB4 QB5 QB6 QB7 QB8 QB9