If c is another common divisor of a and b, then c also divides as + bt

If a and b are coprime and a divides the product bc, then a divides c. This can be viewed as a generalization of Euclid's lemma.

In particular, the third anathema reads: " If anyone divides in the one Christ the hypostases after the union, joining them only by a conjunction of dignity or authority or power, and not rather by a coming together in a union by nature, let him be anathema.

If R is a commutative ring, and a and b are in R, then an element d of R is called a common divisor of a and b if it divides both a and b ( that is, if there are elements x and y in R such that d · x = a and d · y = b ).

If d is a common divisor of a and b, and every common divisor of a and b divides d, then d is called a greatest common divisor of a and b.

If the field of scalars of the vector space has characteristic p, and if p divides the order of the group, then this is called modular representation theory ; this special case has very different properties.

If any of them divides evenly, write 2 at the top of the table and the result of division by 2 of each factor in the space to the right of each factor and below the 2.

# If p is an odd prime, then any prime q that divides 2 < sup > p </ sup > − 1 must be 1 plus a multiple of 2p.

#* Proof: If q divides 2 < sup > p </ sup > − 1 then 2 < sup > p </ sup > ≡ 1 ( mod q ).

# If p is an odd prime, then any prime q that divides must be congruent to ± 1 ( mod 8 ).

A number a is a root of P if and only if the polynomial x − a ( of degree one in x ) divides P. It may happen that x − a divides P more than once: if ( x − a )< sup > 2 </ sup > divides P then a is called a multiple root of P, and otherwise a is called a simple root of P. If P is a nonzero polynomial, there is a highest power m such that ( x − a )< sup > m </ sup > divides P, which is called the multiplicity of the root a in P. When P is the zero polynomial, the corresponding polynomial equation is trivial, and this case is usually excluded when considering roots: with the above definitions every number would be a root of the zero polynomial, with undefined ( or infinite ) multiplicity.

If R is an integral domain and f and g are polynomials in R, it is said that f divides g or f is a divisor of g if there exists a polynomial q in R such that f q = g. One can show that every zero gives rise to a linear divisor, or more formally, if f is a polynomial in R and r is an element of R such that f ( r ) = 0, then the polynomial ( X − r ) divides f. The converse is also true.

If the automatic is placed in reverse or first gear, the transmission divides the torque 50-50 to both front and rear wheels.

Fermat's little theorem states that if p is prime and a is coprime to p, then a < sup > p − 1 </ sup > − 1 is divisible by p. If a composite integer x is coprime to an integer a > 1 and x divides a < sup > x − 1 </ sup > − 1, then x is called a Fermat pseudoprime to base a.

If n is not a prime power, then every Sylow subgroup is proper, and, by Sylow's Third Theorem, we know that the number of Sylow p-subgroups of a group of order n is equal to 1 modulo p and divides n. Since 1 is the only such number, the Sylow p-subgroup is unique, and therefore it is normal.

If one divides a change in proper distance by the interval of cosmological time where the change was measured ( or takes the derivative of proper distance with respect to cosmological time ) and calls this a " velocity ", then the resulting " velocities " of galaxies or quasars can be above the speed of light, c. This apparent superluminal expansion is not in conflict with special or general relativity, and is a consequence of the particular definitions used in cosmology.

If the internal bisector of angle A in triangle ABC has length and if this bisector divides the side opposite A into segments of lengths m and n, then

If these groups have N < sub > p </ sub > and N < sub > q </ sub > elements, respectively, then for any point P on the original curve, by Lagrange's theorem, k > 0 is minimal such that on the curve modulo p implies that k divides N < sub > p </ sub >; moreover,.

# If it is even, the last part is " father ," and one divides by 2.

If man is actually the product of his environment and if science can discover the laws of human nature and the ways in which environment determines what people do, then someone -- a someone probably standing outside traditional systems of values -- can turn around and develop completely efficient means for controlling people.

If private brand competition hasn't been felt in your product field as yet, have you thought how you will cope with it if and when it does appear??

If you have a higher-quality product, how can you make it stand out -- justify its premium price -- without the spoken word??

If competition beats you to it, this exciting new product era can have real headaches in store.

If T is the total `` length '' of the process, its feed state may be denoted by a vector p(T) and the product state by p(Q).

The tensor product X ⊗ Y from X and Y is a K-vector space Z with a bilinear function T: X × Y → Z which has the following universal property: If T ′: X × Y → Z ′ is any bilinear function into a K-vector space Z ′, then only one linear function f: Z → Z ′ with exists.

If the norm of a Banach space satisfies this identity, the associated inner product which makes it into a Hilbert space is given by the polarization identity.

If products designed for the new standard can receive, read, view or play older standards or formats, then the product is said to be backward-compatible ; examples of such a standard include data formats and communication protocols.

If a system is composed of two subsystems described in V and W respectively, then the Hilbert space of the entire system is the tensor product of the two spaces.

If is a ket in V and is a ket in W, the direct product of the two kets is a ket in.

If the dot product is zero, the two vectors are said to be orthogonal to each other.

If the earlier completion procedure is applied to a normed vector space, the result is a Banach space containing the original space as a dense subspace, and if it is applied to an inner product space, the result is a Hilbert space containing the original space as a dense subspace.

If we view the DFT as just a coordinate transformation which simply specifies the components of a vector in a new coordinate system, then the above is just the statement that the dot product of two vectors is preserved under a unitary DFT transformation.

* If we think of as the set of real numbers, then the direct product is precisely just the cartesian product,.

* If we think of as the group of real numbers under addition, then the direct product still consists of.

* If we think of as the ring of real numbers, then the direct product again consists of.

If one alters a Euclidean space so that its inner product becomes negative in one or more directions, then the result is a pseudo-Euclidean space.

If two numbers have no prime factors in common, their greatest common divisor is 1 ( obtained here as an instance of the empty product ), in other words they are coprime.

Tensor products: If C denotes the category of vector spaces over a fixed field, with linear maps as morphisms, then the tensor product defines a functor C × C → C which is covariant in both arguments.

A former senior product manager at Master Lock, a trigger lock manufacturer, was quoted as saying “ If you put a trigger lock on any loaded gun, you are making the gun more dangerous .” Critics also point out that a trigger lock will increase the time it takes a gun owner to respond to a self-defense emergency.

If a siRNA is designed to match the RNA copied from a faulty gene, then the abnormal protein product of that gene will not be produced.

If the magnetic field is applied by a solenoid, the sensor output is proportional to the product of the current through the solenoid and the sensor voltage.