<p>For each positive integer, <img alt="$n$" height="10" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="13" /> let <img alt="$a_n$" height="13" src="https://latex.artofproblemsolving.com/6/f/1/6f1a2b6b7c193f9ba2b2134a59f1da5addcfbc98.png" width="20" /> be the least positive integer multiple of <img alt="$23$" height="14" src="https://latex.artofproblemsolving.com/9/a/1/9a1b2497e7b125cd721bab4c6e2385f954fe35cb.png" width="20" /> such that <img alt="$a_n \equiv 1 \pmod{2^n}.$" height="20" src="https://latex.artofproblemsolving.com/1/d/c/1dcc0454651db1fc4bcf71044a46d1a124125f48.png" width="140" /> Find the number of positive integers <img alt="$n$" height="10" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="13" /> less than or equal to <img alt="$1000$" height="14" src="https://latex.artofproblemsolving.com/6/e/e/6ee927e1332358c96c62c277441c907c4f51057f.png" width="38" /> that satisfy <img alt="$a_n = a_{n+1}.$" height="14" src="https://latex.artofproblemsolving.com/f/a/0/fa0137ab7fd126ff25aa6b5f674174c3a39cd3c4.png" width="85" /></p>

Question No: 446

For each positive integer, $n$ let $a_n$ be the least positive integer multiple of $23$ such that $a_n \equiv 1 \pmod{2^n}.$ Find the number of positive integers $n$ less than or equal to $1000$ that satisfy $a_n = a_{n+1}.$

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Tags: Number Theory

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