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The greatest common factor (GCF) is the largest positive divisor shared by all of the compared positive integers. It is the maximum value in the set of their common factors.

Conditions: At least two positive integers are compared.; Positive divisors are compared.

The span of two nonzero, non-collinear vectors is the entire 2D plane. This means that by taking all possible linear combinations of these two vectors, you can reach any point in the plane.

Conditions: The two vectors are nonzero.; The two vectors are non-collinear (they do not lie on the same line).

The determinant represents the scalar factor by which any region's area changes under the transformation. For example, if a unit square becomes a rectangle with an area of 6, the determinant is 6.

Conditions: Linear transformation in 2D space; Measurable planar regions

The first three running averages are 55, 60, and 55. The first average is just the first observation (55).

Conditions: Observations are 55, 65, and 45.; The sample mean is calculated cumulatively.

For any set of distinct positive numbers, the harmonic mean is strictly the smallest, followed by the geometric mean, and the arithmetic mean is the largest. Equality holds only when all input numbers are identical.

Conditions: Inputs are positive real numbers; Comparing H, G, and A means

The constant multiple rule is a special case of the product rule where one factor is a constant $c$. Geometrically, multiplying a function by a constant scales its output heights (and thus its area strips) by $c$.

Conditions: $c$ is a constant.; The function $f$ is differentiable.

If the columns of a transformation matrix are linearly dependent, it means the transformation collapses the space into a lower dimension. For example, a 2D plane might be squashed into a 1D line or a single point.

Conditions: The matrix represents a linear transformation.; Columns are linearly dependent.

For a uniform partition of the radius $[0, R]$ into $n$ subintervals, the difference between the upper sum (overestimation) and the lower sum (underestimation) is exactly $\frac{2\pi R^2}{n}$. This demonstrates that the error decreases inversely with the number of partitions $n$.

Conditions: Uniform partition of radius $R$ into $n$ steps; Linear circumference function $C(r) = 2\pi r$

The matrix representation is $\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$. In this transformation, the standard basis vector $\hat{i}$ remains fixed at $[1, 0]^T$, while $\hat{j}$ slides diagonally to $[1, 1]^T$.

Conditions: Horizontal shear keeping $\hat{i}$ fixed.; $\hat{j}$ slides diagonally to $[1, 1]^T$.

When a third vector pokes out of the plane at an angle, scaling it allows the entire 2D sheet to slide up and down through space. With three freely varying scalars ($a\vec{v} + b\vec{w} + c\vec{u}$), the span now encompasses every conceivable point in the 3D volume.

Conditions: The first two vectors span a plane in 3D space.; The third vector is not on that plane (it pokes out at an angle).; The scalars $a$, $b$, and $c$ vary freely.

Trivial factors are the readily available factors 1 and the integer itself. For any positive integer greater than 1, these two factors are distinct.

Conditions: The target is a positive integer greater than 1.

In the step $45 = 10 \cdot q + r$, $q$ represents the quotient, which counts how many whole times the smaller number (10) fits into the larger number (45). $r$ represents the remainder, which is the leftover amount after subtracting those whole multiples.

Conditions: The equation is part of the division-with-remainder step of the Euclidean algorithm.; The inputs are positive integers.; The remainder satisfies $0 \le r < 10$.