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Olympiad workbook
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IOQM
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Basics of Mathematics

Olympiad workbook · IOQM · Basics of Mathematics

Let N denote the number of all natural numbers n such that n is...

Let NN denote the number of all natural numbers nn such that nn is divisible by a prime p>np>\sqrt{n} and p<20\mathrm{p} < 20. What is the value of N ?

Correct answer

69

Step-by-step solution

Since P>n\mathrm{P}>\sqrt{\mathrm{n}} and P>20\mathrm{P}>20 Hence for the times from 2 to 20 we calculate possibilities that that n<P2\mathrm{n} < \mathrm{P}^2 (Prime No.Natural Number n2233,655,10,15,2077,14,21,28,35,421111,22,33,44,55,66,77,88,99,1101313,26,39,52,65,78,91,104,117,130,143,1561717,34,51,68,85,102,119,136,153,170,187,204,221,238,255,2721919,38,57,76,95,114,133,152,171,190,209,228,247,266,285,304,323,342\begin{array}{|c|l|} \hline \text{Prime No.} & \text{Natural Number } n \\ \hline 2 & 2 \\ \hline 3 & 3,6 \\ \hline 5 & 5,10,15,20 \\ \hline 7 & 7,14,21,28,35,42 \\ \hline 11 & 11,22,33,44,55,66,77,88,99,110 \\ \hline 13 & 13,26,39,52,65,78,91,104,117,130,143,156 \\ \hline 17 & 17,34,51,68,85,102,119,136,153,170, \\ & 187,204,221,238,255,272 \\ \hline 19 & 19,38,57,76,95,114,133,152,171,190, \\ & 209,228,247,266,285,304,323,342 \\ \hline \end{array}) Total numbers = 69
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