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Omni-MATH / P.J. starts with m=500 and chooses a positive integer n with 1≤n≤499. He applies the following algorithm to m and n: P.J. sets r equal to the remainder when m is divided by n. If …

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problem

P.J. starts with m=500m=500 and chooses a positive integer nn with 1≤n≤4991 \leq n \leq 499. He applies the following algorithm to mm and nn: P.J. sets rr equal to the remainder when mm is divided by nn. If r=0r=0, P.J. sets s=0s=0. If r>0r>0, P.J. sets ss equal to the remainder when nn is divided by rr. If s=0s=0, P.J. sets t=0t=0. If s>0s>0, P.J. sets tt equal to the remainder when rr is divided by ss. For how many of the positive integers nn with 1≤n≤4991 \leq n \leq 499 does P.J.'s algorithm give 1≤r≤151 \leq r \leq 15 and 2≤s≤92 \leq s \leq 9 and t=0t=0?
Plain-text mathematical notation (without MathML)
P.J. starts with m=500 and chooses a positive integer n with 1≤n≤499. He applies the following algorithm to m and n: P.J. sets r equal to the remainder when m is divided by n. If r=0, P.J. sets s=0. If r>0, P.J. sets s equal to the remainder when n is divided by r. If s=0, P.J. sets t=0. If s>0, P.J. sets t equal to the remainder when r is divided by s. For how many of the positive integers n with 1≤n≤499 does P.J.'s algorithm give 1≤r≤15 and 2≤s≤9 and t=0?
Original LaTeX notation
P.J. starts with \(m=500\) and chooses a positive integer \(n\) with \(1 \leq n \leq 499\). He applies the following algorithm to \(m\) and \(n\): P.J. sets \(r\) equal to the remainder when \(m\) is divided by \(n\). If \(r=0\), P.J. sets \(s=0\). If \(r>0\), P.J. sets \(s\) equal to the remainder when \(n\) is divided by \(r\). If \(s=0\), P.J. sets \(t=0\). If \(s>0\), P.J. sets \(t\) equal to the remainder when \(r\) is divided by \(s\). For how many of the positive integers \(n\) with \(1 \leq n \leq 499\) does P.J.'s algorithm give \(1 \leq r \leq 15\) and \(2 \leq s \leq 9\) and \(t=0\)?

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Source and history

task_id
72450be5-1bff-54f6-96b3-06d3d75a9c9c
task_key
test--72450be5-1bff-54f6-96b3-06d3d75a9c9c
task_revision_id
1
config / split
/ test

Official source

initial import