Question

How do I find the correct variable for N? (It won't be negative I think)

Answer 1

`25`

Explanation:

`\mathrm{For\ the\ set\ to\ be\ the\ sets\ of\ natural\ numbers,\ (n)/(5)\ and\ √(n+144)\ must\ be\ a}\\\mathrm{positive\ integer.}\\\mathrm{√(n+144)\ is\ a\ natural\ number\ when\ (n+144)\ is\ a\ perfect\ square.}\\\mathrm{And\ (n)/(5)\ is\ a\ natural\ number\ if\ and\ only\ if\ n\ is\ exactly\ divisible\ by\ 5.}\\\mathrm{So\ we\ need\ to\ find\ the\ smallest\ positive\ integer\ which\ would\ make\ both\ of\ these}\\\mathrm{cases\ true.}`

`\mathrm{Now\ we\ know\ that\ 144\ is\ a\ perfect\ square.\ The\ most\ closed\ perfect\ square\ to}\\\mathrm{144\ are\ 121\ and\ 169.}\\\mathrm{We\ are\ taking\ these\ closest\ perfect\ squares\ to\ 144\ because\ we\ want\ the\ value\ of}\\\mathrm{n\ to\ be\ as\ small\ as\ possible.}\\\mathrm{Now\ if\ (n+144)=121,\ n=23\ (which\ is\ not\ divisible\ by\ 5).}\\\mathrm{if\ (n+144)=169,\ n=25(which\ is\ divisible\ by\ 5).}`

`\mathrm{Notice,\ when\ n=25,\ 34+n,\ (n)/(5)\ and\ √(n+144)\ are\ all\ natural\ numbers.}\\\therefore\ \mathrm{n}=25.`