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How many positive integers less than 60 have an even number of positive divisors?

 Jun 30, 2019
 #1
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If I remember correctly, ALL numbers between 2 and 59 have an EVEN numbers of positive divisors, EXCEPT the Perfect Squares, or 4, 9, 16, 25, 36 and 49, which have 3 divisors each. So:

59 - 2 + 1 =58 - 6 perfect squares =52 such numbers that have EVEN divisors.

 Jun 30, 2019
 #2
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+2

How many positive integers less than \(60\) have an even number of positive divisors?

 

\(\begin{array}{|r|r|l|} \hline n & \text{the number of divisors of }n \\ \hline 1 & 1 & {\color{red}odd} \\\\ 2&2& {\color{green}even} \\ 3&2& {\color{green}even} \\ 4&3& {\color{red}odd} \\ \\ 5&2& {\color{green}even} \\ 6&4& {\color{green}even} \\ 7&2& {\color{green}even} \\ 8&4& {\color{green}even} \\ 9&3& {\color{red}odd} \\ \\ 10&4& {\color{green}even} \\ 11&2& {\color{green}even} \\ 12&6& {\color{green}even} \\ 13&2& {\color{green}even} \\ 14&4& {\color{green}even} \\ 15&4& {\color{green}even} \\ 16&5& {\color{red}odd} \\\\ 17&2& {\color{green}even} \\ 18&6& {\color{green}even} \\ 19&2& {\color{green}even} \\ 20&6& {\color{green}even} \\ 21&4& {\color{green}even} \\ 22&4& {\color{green}even} \\ 23&2& {\color{green}even} \\ 24&8& {\color{green}even} \\ 25&3& {\color{red}odd} \\ \\ 26&4& {\color{green}even} \\ 27&4& {\color{green}even} \\ 28&6& {\color{green}even} \\ 29&2& {\color{green}even} \\ 30&8& {\color{green}even} \\ 31&2& {\color{green}even} \\ 32&6& {\color{green}even} \\ 33&4& {\color{green}even} \\ 34&4& {\color{green}even} \\ 35&4& {\color{green}even} \\ 36&9& {\color{red}odd} \\ \\ 37&2& {\color{green}even} \\ 38&4& {\color{green}even} \\ 39&4& {\color{green}even} \\ 40&8& {\color{green}even} \\ 41&2& {\color{green}even} \\ 42&8& {\color{green}even} \\ 43&2& {\color{green}even} \\ 44&6& {\color{green}even} \\ 45&6& {\color{green}even} \\ 46&4& {\color{green}even} \\ 47&2& {\color{green}even} \\ 48&10& {\color{green}even} \\ 49&3& {\color{red}odd} \\\\ 50&6& {\color{green}even} \\ 51&4& {\color{green}even} \\ 52&6& {\color{green}even} \\ 53&2& {\color{green}even} \\ 54&8& {\color{green}even} \\ 55&4& {\color{green}even} \\ 56&8& {\color{green}even} \\ 57&4& {\color{green}even} \\ 58&4& {\color{green}even} \\ 59&2& {\color{green}even} \\ \hline && \text{sum = } \color{green}52 \\ \hline \end{array} \)

 

laugh

 Jul 1, 2019

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