Property of numbers

Example:

if d>0 and 0<1-c/d <1, which of the followng must be true?

I. c>0

II. c/d <1

III. c^2 + d^2 >1

If x, y, and z are positive integers such that x is a factor of y, and x is a multiple of z, which of the following is NOT an integer?
a) x+z
Z
b) y+z
x
c) x+y
Z
d) xy/
z
e) yz/x
x

In example I statement II is true since from left inequality we have c/d<1 so you can stop right away here but if you want to check other cases take c=3,d=1 so 1-c/d=-2 but left ineq.does not hold 0<-2(nonsense)
and for last case take c=0,1 and d=0,2 which their squares if you add them up are less than 1
Note: a logical question is: are there any numbers 0<x,y<1 such that if you square them and add them up their sum could be >=1. The answer is definitely no. A simple way to remember this is to consider the unit circle whose equation is x^2+y^2 = 1. (all pts with the property -1<x<1 or 0<x<1 are inside this circle so x^2+y^2 <1
thanx

What are the OAs to the questions mentioned above?

For first: My answer is I and II.

Thanks
Mankey

For 1,

subtract -1 from both sides

-1<-c/d<0. Multiply both sides by -1; 1>c/d>0

This tells us that c/d<1 and c and d are positive fractions.
That answers i and ii.

iii need be true always. Take c=0.2 and d=0.3. c/d<1 and the sum of the squares is less than 1

For 2, we can rephrase the question as
y/x = integer and x/z = integer

So, a,b,c are all sum of some positive integers. We can rule them out

Let us try d
xy/z = x*x*(integer)/z.
Let us try e
yz/x = x*(integer)*z/x. Cancel the x's, we have z*integer. So D is not an integer