A regular polygon is defined as a polygon with all sides and angles equal, and it cannot be concave; it must be convex.

Hybrid sets can include duplicate elements, especially when they are formed from multiple sources.

A circle is defined as a round shape with no corners.

In a square, all angles are right angles (90 degrees), not acute.

In a geometric progression, the common ratio cannot be zero, as it would invalidate the progression.

The sum of the terms in a harmonic progression is not necessarily an integer, as the terms can be fractions.

The sum of the first n terms of a harmonic progression diverges as n approaches infinity, hence it is not finite.

The last statement is incorrect as it does not follow the properties of logarithms.

Modular arithmetic is commutative for addition and multiplication, hence non-commutativity is not a property.

Polygons are defined as shapes made up of straight line segments, so having curved sides disqualifies a shape from being a polygon.

Not all triangles have all angles acute; for example, a right triangle has one angle that is 90 degrees.

'Algorithm' is a mathematical concept, while the others are literary devices used in writing.

'22' cannot represent 10 in any base since it would require base 3 or higher.

The passage does not mention limited funding as a challenge, focusing instead on other obstacles faced by the researchers.

A factor is defined as a number that divides another number without leaving a remainder.

The quadratic x^2 - 5x + 6 factors to (x - 2)(x - 3).

All the listed properties are correct and fundamental to logarithmic functions.

The decimal number 15 is represented as '1111' in binary.

Using the power of a product property, (a*b)^n = a^n * b^n, we get (x^3)^2 * (y^2)^2 = x^6 * y^4.

Using the property of logarithms, log_2(8) + log_2(4) = log_2(8*4) = log_2(32) = log_2(16).

log_5(25) = 2 and log_5(5) = 1, thus log_5(25) - log_5(5) = 2 - 1 = 1.

The GCD of 48 and 180 is 12, as it is the largest number that divides both without a remainder.

Using the power of a power property, (a^m)^n = a^(m*n), we get (2^3)^2 = 2^(3*2) = 2^6.

The polynomial can be factored as (x + 2)^2, indicating it has one real root with multiplicity 2.

Using the property of exponents, we have 2^(x+1 - (x-1)) = 2^(x+1-x+1) = 2^2.

Since 16 can be expressed as 4^2, we have 4^(x+1) = 4^2, leading to x + 1 = 2, thus x = 1.

Since 0.01 is 10^-2, log_10(0.01) = -2.

Logarithm of zero is undefined, hence log_5(0) is the correct answer.

12 is the smallest number that is a multiple of both 4 and 6.

The LCM of 8 and 12 is 24, which is a multiple of both.