3rd Edition, Guide 4, Hard questions pp.179, #9

[john.joseph.hall]

If (x^2)/4 is an integer greater than 50, then what is the smallest possible value for x^2?

I claim 204. The guide claims: 256- why must x itself be an integer? Why can x not be: 2*sqr(51)?

[jen]

Hi there! Great question.

x^2 must be an integer because we are told that (x^2)/4 is an integer. In order for (x^2)/4 to be an integer, x^2 needs to have prime factors of 2 and 2, and possibly some other integers. But x^2 definitely has to be an integer for (x^2)/4 to be one.

Note that the problem doesn't ask for the smallest possible x, in which case, THAT would not have to be an integer (for instance 2rad2 would work, since when you square it, you get 8, and 8/4 is an integer).

However, the problem asks for the smallest possible value of X SQUARED, not x.

Sincerely,
Jen

[manigraja100]

Hi jen,
Even after i read your explanation,i couldn't understand why x^2 can't be 204. given condition does not say that x^2 has to be an integer.it just says that x^2/4 has to be an integer.so,
204/4=51 suits the condition. where am i getting it wrong ?

[tommywallach]

Hey Mani,

Yeah, Jen is wrong here. (It happens! She's right for 8 hours a day 364 days a year!). John's explanation is correct. : )

-t