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In the Euclidean algorithm, the previous divisor becomes the new dividend (placed on the left side of the equation), and the previous remainder becomes the new divisor (placed on the right side). This recursive shift carries the numbers forward so that each step divides the former divisor by the former remainder, continuing until a remainder of zero is reached.

Conditions: The algorithm is applied to positive integers.; The previous remainder is not zero.; The process follows the standard division-with-remainder format $a = b \cdot q + r$.

Matrix-vector multiplication in 3D works by scaling the columns of the matrix by the corresponding coordinates of the input vector and summing the results. The coordinates $(x, y, z)$ act as scalar multipliers for the transformed basis vectors (the columns), leveraging the linearity property that preserves addition and scalar multiplication.

Conditions: The matrix is a 3x3 transformation matrix.; The input vector has coordinates $(x, y, z)$.; The transformation is linear.

Multiplying two 3x3 matrices represents applying their respective spatial transformations sequentially. The rightmost matrix acts first on the initial space, and the leftmost matrix acts on the already transformed result.

Conditions: Both matrices are 3x3 transformation matrices.; The operation is standard matrix multiplication.

By tracking a small input change $dx$ through nested mappings $x \to h(x) \to g(h(x))$. For $x=1.5$, $h=x^2$, and $g=\sin$, the inner rate is $h'(1.5)=3$, scaling $dx$ to $dh \approx 3dx$.

Conditions: Inner function $h(x)$ is differentiable at $x$; Outer function $g(u)$ is differentiable at $u=h(x)$; Example uses $h(x)=x^2$ and $g(u)=\sin u$ near $x=1.5$

The length is multiplied by the absolute value of the scalar $|c|$. Specifically, $0 < |c| < 1$ shortens the vector, $|c| > 1$ stretches it, and $c=0$ yields the zero vector.

Conditions: Scalar multiplication operation; Considering magnitude/length changes

To calculate the unique permutations, first determine the total arrangements assuming all letters are distinct, which is $7!$ for the 7-letter word RATTATA. Then, adjust for duplicates by dividing by the factorial of the count of each repeating letter group.

Conditions: The word contains repeated letters.; All positions in the permutation are filled.

Knowing the greatest common factor (GCF) of 12 and 42, which is 6, allows you to divide both the numerator and the denominator by 6 directly. This immediately yields the simplest form $2/7$, bypassing intermediate steps like dividing by 2 first.

Conditions: The fraction is $12/42$.; The GCF of 12 and 42 is known to be 6.

Geometrically, adding two vectors involves placing the tail of the second vector at the head of the first; the sum is the vector drawn from the start of the first to the finish of the second.

Conditions: Two vectors available for addition; Euclidean geometric representation

When $A\vec{x}=\vec{b}$ has no exact solution, the least-squares solution $\vec{x}^*$ is defined as the vector that minimizes the Euclidean norm of the residual, $\|\vec{b} - A\vec{x}^*\|$. Geometrically, this means choosing $\vec{x}^*$ such that $A\vec{x}^*$ is the closest possible vector to $\vec{b}$ within the column space $C(A)$.

Conditions: The system $A\vec{x}=\vec{b}$ is inconsistent (no exact solution exists); Distance is measured by the standard Euclidean norm

To construct the 3x3 matrix, observe where the three standard basis vectors ($\hat{i}$, $\hat{j}$, and $\hat{k}$) land after the transformation. Record the final coordinates of these transformed vectors as column vectors.

Conditions: Working in three-dimensional Cartesian space.; The transformation is linear.

The intuitive concept says $f(x)$ 'gets closer' to $L$ as $x$ 'gets closer' to $a$. The epsilon-delta definition formalizes this by replacing vague 'closeness' with precise quantitative tolerances: $\epsilon$ measures output closeness to $L$, and $\delta$ measures input closeness to $a$.

Conditions: The intuitive notion of 'approaching' is qualitative.; The formal definition introduces universal and existential quantifiers over positive real numbers.

The fundamental theorem relates accumulation and rate of change by stating that if $F'(x) = f(x)$, then the accumulated area $A(x)$ from a fixed point $a$ to $x$ is given by $A(x) = F(x) - F(a)$. It shows that differentiation and integration are inverse processes: the derivative of the accumulation function $A(x)$ recovers the integrand $f(x)$, and antiderivatives allow the evaluation of definite integrals.

Conditions: The function $f$ is continuous.; $F$ is an antiderivative of $f$ (i.e., $F' = f$).