Number Properties Guide : Hard Practice : Q9 pg. 179

[xerocoool]

Hi,

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

Please let me know if it's the right way of approaching such types of sums .

Solution:
Step1: Since x is divisible by 4 : x has to be multiple of 4 (4,8,12,16,20..)
Step2: x being multiple of 4 : x^2 is multiple of 4 (i.e 16,64,144,256,400..)
Step3: 256/4 = 64

In addition, given the case if x is not an integer. How does one arrive at the answer 204.

The question states x^2/4 is an integer greater than 50 ; by property of divisibility "On dividing an integer by another integer if the result/quotient is an integer then integer 1 is divisible by integer 2 ; does this not automatically make x or x^2 an integer value ? (as the problem mentions the result is an integer)

Thanks
-X

[tommywallach]

Hey Xero,

You go wrong right up top. The question never says that x is a multiple of 4. It says that x^2 is a multiple of 4. That's different. For example:

2 --> not a multiple of 4

2^2 --> is a multiple of 4

So you're starting with a false premise right there. You should just look at the inequality here:

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

x^2/4 > 50

x^2 > 200

We already know that x is an integer. So all we need is whatever the first perfect square is that's above 200 (and is also divisible by 4. 14^2 = 196. That's not big enough. 15^2 = 225. That's big enough, but it's not divisible by 4. 16^2 = 256. That's big enough and divisible by 4. Voila!

-t