The area of a circle is given by the formula A = πr² where r is the radius of the circle. The radius of a circle is its diameter divided by two so A = π(\( \frac{d}{2} \))². If the diameter of the logo increases by 65% the radius (and, consequently, the total area) increases by \( \frac{65\text{%}}{2} \) = 32\(\frac{1}{2}\)%

The first few multiples of 5 are [5, 10, 15, 20, 25, 30, 35, 40, 45, 50] and the first few multiples of 9 are [9, 18, 27, 36, 45, 54, 63, 72, 81, 90]. The first few multiples they share are [45, 90] making 45 the smallest multiple 5 and 9 have in common.

First, solve for \( \left|z - 5\right| \):

\( \left|z - 5\right| \) + 1 = 4
\( \left|z - 5\right| \) = 4 - 1
\( \left|z - 5\right| \) = 3

The value inside the absolute value brackets can be either positive or negative so (z - 5) must equal + 3 or -3 for \( \left|z - 5\right| \) to equal 3:

So, z = 2 or z = 8.

The greatest common factor (GCF) is the greatest factor that divides two integers.

A factorial is the product of an integer and all the positive integers below it. To solve a fraction featuring factorials, expand the factorials and cancel out like numbers:

\( \frac{3!}{2!} \)
\( \frac{3 \times 2 \times 1}{2 \times 1} \)
\( \frac{3}{1} \)
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