90 lines
5.2 KiB
HTML
90 lines
5.2 KiB
HTML
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<h2>Largest integer divisible by two primes</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 347</h3><span style="width:300px;color:#666;">Published on Saturday, 3rd September 2011, 04:00 pm; Solved by 1238</span></div>
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The largest integer <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> 100 that is only divisible by both the primes 2 and 3 is 96, as 96=32*3=2<sup>5</sup>*3.
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For two <i>distinct</i> primes p and q let M(p,q,N) be the largest positive integer <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' />N only divisible
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by both p and q and M(p,q,N)=0 if such a positive integer does not exist.
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E.g. M(2,3,100)=96.<BR/>
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M(3,5,100)=75 and not 90 because 90 is divisible by 2 ,3 and 5.<BR/>
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Also M(2,73,100)=0 because there does not exist a positive integer <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> 100 that is divisible by both 2 and 73.
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Let S(N) be the sum of all distinct M(p,q,N).
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S(100)=2262.
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Find S(10 000 000).
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