Numbers Properties Guide : Doubts

[xerocoool]

Hi,

Any Help will be greatly appreciated :)

Q.8) Pg.166 Medium Problem Set (difficulty in understanding the logic behind the sum)

Q.9) Pg.170 : 2nd Edition : Hard Problem Set
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 ?

Although I have got the right answer, 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. Could you please let me know how does one get 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)

I am not quite sure how to treat x^n terms/divisible by something, facing similar problem in Q18 Pg.171 (could you please explain this sum too.)

Q.2) Pg 165 : Medium Problem Set

x/100 is an integer
Quantity A: x^3
Quantity B: x^2+100

The guide states "Question related to divisibility are only interested in positive integers". Should I be keeping in mind negative multiples/factors as well ?

Q17) Pg.171 : Hard Problem Set

b,c,d are consecutive even integers such that 2<b<c<d. What is the largest positive integer that must be a divisor of bcd.

Soln. I believe, the question is asking for the largest positive factor of bcd
Using LCM and the condition 2<4<6<8
4: 2 x 2
6: 2 x 3
8: 2 x 2 x 2
LCM : 2x3x2x2x2 (basic building blocks for the numbers after prime factorization of 4,6,8)
And hence 48

Was unable to get the explanation given in the book, I hope the above method is right.

Apologies for the long post.

Thanks,
-X

[tommywallach]

Hey X,

You're going to hate me, but could you please post these as separate individual questions? I know it's annoying, but you have to understand that this won't be helpful to other people like this, because they won't know what you meant by doubts, so they won't click it. Instead, do separate posts with the book name/page number/question #, so that if other people have similar questions, they'll know to check that thread. Make sense?

-t

[xerocoool]

sure no problem.

[tommywallach]

Thank you!

-t