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(This article was first published on <strong><a href="https://xianblog.wordpress.com/2018/04/01/le-monde-puzzle-1047/"> R – Xi'an's Og</a></strong>, and kindly contributed to <a rel="nofollow" href="https://www.r-bloggers.com/">R-bloggers)</a> </div>
<p><strong><a rel="nofollow" href="https://xianblog.files.wordpress.com/2018/03/sieglemond.jpg" target="_blank"><img data-orig-file="https://xianblog.files.wordpress.com/2018/03/sieglemond.jpg" data-orig-size="660,281" data-medium-file="https://xianblog.files.wordpress.com/2018/03/sieglemond.jpg?w=300" data-attachment-id="35577" data-image-meta="{" data-image-description="" src="https://xianblog.files.wordpress.com/2018/03/sieglemond.jpg?w=450&h=192&fit=450%2C192&resize=450%2C192" data-permalink="https://xianblog.wordpress.com/2018/03/12/le-monde-puzzle-1044/sieglemond/" data-large-file="https://xianblog.files.wordpress.com/2018/03/sieglemond.jpg?w=450&amp;h=192&amp;fit=450%2C192&amp;resize=450%2C192" alt="" data-image-title="sieglemond" sizes="(max-width: 450px) 100vw, 450px" width="450" data-comments-opened="1" class="wp-image-35577" data-recalc-dims="1" height="192"></a>A</strong>n arithmetic <a rel="nofollow" href="https://xianblog.wordpress.com/2011/09/03/le-monde-puzzle-website/" title="Le Monde puzzle [website]" target="_blank">Le Monde mathematical puzzle</a>:</p>
<blockquote>
<p><em>A magical integer m is such that the remainder of the division of any prime number p by m is either a prime number or 1. What is the unique magical integer between 25 and 100? And is there any less than 25?</em></p>
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<p>The question is dead easy to code</p>
<pre>primz=c(1,generate_primes(2,1e6))
for (y in 25:10000)
  if (min((primz[primz>y]%%y)%in%primz)==1) print(y)
</pre>
<p>and return m=30 as the only solution. Bon sang but of course!, since 30=2x3x5… (Actually, the result follows by dividing the quotient of the division of a prime number by 2 by 3 and then the resulting quotient by 5: all possible cases produce a remainder that is a prime number.) For the second question, the same code returns 2,3,4,6,8,12,18,24 as further solutions. There is no solution beyond 30.</p>
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