Problem 5
A counterexample could be a number like 6. Divide it over 2 and we get 6/2 = 3, which is a whole number. This shows 6 is a multiple of 2. However, 6/4 = 1.5 is not a whole number, so 6 is not divisible by 4. The overall claim is false.
Problem 6
Let's say we are subtracting a positive number and a negative number. For instance, let's say we're subtracting 10 and -7
So,
10 minus -7 = 10 - (-7) = 10+7 = 17
When subtracting a negative, the two minus signs cancel to form a plus sign. The result we get is 17, which is larger than the greater number 10. So this is one counterexample that proves the claim to be false.
Problem 7
Start with a circle. The circle can be any size. Plot four random points on the circle. The points can be anywhere you want. Let's label those points A,B,C,D. Form quadrilateral ABCD.
Now let's say we pick point A and pull it off the circle and pull it outside the circle. Points B,C and D remain on the circle. This is one example where we cannot draw a circle through this new arrangement of points. The circle goes through B,C, and D just fine; however, it doesn't go through point A. If you try to attempt to get the circle to go through A, then you'll have to pick B,C or D to ignore. In other words, you'll only be able to pick three points that the circle can go through. Therefore, a circle cannot be drawn as described for every quadrilateral.
Note: Any quadrilateral that has all four points on the same circle is known as a cyclic quadrilateral.
