* Every pair of congruence relations for an unknown integer x, of the form x ≡ k ( mod a ) and x ≡ l ( mod b ), has a solution, as stated by the Chinese remainder theorem ; in fact the solutions are described by a single congruence relation modulo ab.

Every real number, whether integer, rational, or irrational, has a unique location on the line.

Every positive integer n > 1 can be represented in exactly one way as a product of prime powers:

: Every even integer greater than 2 can be expressed as the sum of two primes.

: Every integer which can be written as the sum of two primes, can also be written as the sum of as many primes as one wishes, until all terms are units.

: Every integer greater than 2 can be written as the sum of three primes.

: Every integer greater than 5 can be written as the sum of three primes.

: Every even integer greater than 2 can be written as the sum of two primes,

Every positive integer appears exactly once somewhere on this list.

* Every root of a monic polynomial whose coefficients are algebraic integers is itself an algebraic integer.

Every memory block in this system has an order, where the order is an integer ranging from 0 to a specified upper limit.

* Every positive integer can be written as the sum of 73 or fewer sixth powers ( see Waring's problem ).

Every positive integer is the sum of at most 37 fifth powers ( see Waring's problem ).

Every integer greater than 96 may be represented as a sum of distinct super-prime numbers.

Every lattice in can be generated from a basis for the vector space by forming all linear combinations with integer coefficients.

* Every positive integer except 1 is a PV number.

In mathematics, an integer-valued polynomial ( also known as a numerical polynomial ) P ( t ) is a polynomial whose value P ( n ) is an integer for every integer n. Every polynomial with integer coefficients is integer-valued, but the converse is not true.

Every positive integer can be expressed as the sum of at most 19 fourth powers ; every sufficiently large integer can be expressed as the sum of at most 16 fourth powers ( see Waring's problem ).

* Every character value is a sum of n m < sup > th </ sup > roots of unity, where n is the degree ( that is, the dimension of the associated vector space ) of the representation with character χ and m is the order of g. In particular, when F is the field of complex numbers, every such character value is an algebraic integer.

A finitely-generated abelian group is indecomposable if and only if it is isomorphic to Z or to a factor group of the form for some prime number p and some positive integer n. Every finitely-generated abelian group is a direct sum of ( finitely many ) indecomposable abelian groups.

Every integer instruction could operate on either 1-byte or 2-byte integers and could access data stored in registers, stored as part of the instruction, stored in memory, or stored in memory and pointed to by addresses in registers.

< li > Every positive definite matrix is invertible and its inverse is also positive definite.

Every aspect of the line matrix printer is designed to deliver higher reliability, fast throughput, and greater resistance to rough handling and hazardous environmental conditions.

Every diagonal matrix is symmetric, since all off-diagonal entries are zero.

Every symmetric matrix is thus, up to choice of an orthonormal basis, a diagonal matrix.

Every real symmetric matrix is Hermitian, and therefore all its eigenvalues are real.

Every orthogonal matrix has determinant either 1 or − 1.

Every symplectic matrix is invertible with the inverse matrix given by

Every Hermitian matrix is a normal matrix, and the finite-dimensional spectral theorem applies.

Every constraint is in turn a pair ( usually represented as a matrix ), where is an-tuple of variables and is an-ary relation on.

Every hypergraph has an incidence matrix where

Every real m-by-n matrix yields a linear map from R < sup > n </ sup > to R < sup > m </ sup >.

* Every finite-dimensional simple algebra over R must be a matrix ring over R, C, or H. Every central simple algebra over R must be a matrix ring over R or H. These results follow from the Frobenius theorem.

* Every finite-dimensional simple algebra over C must be a matrix ring over C and hence every central simple algebra over C must be a matrix ring over C.

* Every finite-dimensional central simple algebra over a finite field must be a matrix ring over that field.

We call a field E a splitting field for A if A ⊗ E is isomorphic to a matrix ring over E. Every finite dimensional CSA has a splitting field: indeed, in the case when A is a division algebra, then a maximal subfield of A is a splitting field.

* Every 4-dimensional central simple algebra over a field F is isomorphic to a quaternion algebra ; in fact, it is either a two-by-two matrix algebra, or a division algebra.

Every object in the drawing can be subjected to arbitrary affine transformations: moving, rotating, scaling, skewing and a configurable matrix.

Every pixel from the secret image is encoded into multiple subpixels in each share image using a matrix to determine the color of the pixels.

Every time I closed my eyes, I saw Gray Eyes rushing at me with a knife.

Every detail in his interpretation has been beautifully thought out, and of these I would especially cite the delicious laendler touch the pianist brings to the fifth variation ( an obvious indication that he is playing with Viennese musicians ), and the gossamer shading throughout.

Every Wednesday night through Oct. 11 there will be an elaborate colored fountain display, with 229 nozzles throwing jets of water up to 130 feet.

Every doll in the house must be going with her ''.

** Tukey's lemma: Every non-empty collection of finite character has a maximal element with respect to inclusion.

Every unit of length has a corresponding unit of area, namely the area of a square with the given side length.

* Revised Code of Washington 9. 12. 010: " Every person who brings on his or her own behalf, or instigates, incites, or encourages another to bring, any false suit at law or in equity in any court of this state, with intent thereby to distress or harass a defendant in the suit, or who serves or sends any paper or document purporting to be or resembling a judicial process, that is not in fact a judicial process, is guilty of a misdemeanor ; and in case the person offending is an attorney, he or she may, in addition thereto be disbarred from practicing law within this state.

Every Boolean algebra ( A, ∧, ∨) gives rise to a ring ( A, +, ·) by defining a + b := ( a ∧ ¬ b ) ∨ ( b ∧ ¬ a ) = ( a ∨ b ) ∧ ¬( a ∧ b ) ( this operation is called symmetric difference in the case of sets and XOR in the case of logic ) and a · b := a ∧ b. The zero element of this ring coincides with the 0 of the Boolean algebra ; the multiplicative identity element of the ring is the 1 of the Boolean algebra.

Every Hilbert space X is a Banach space because, by definition, a Hilbert space is complete with respect to the norm associated with its inner product, where a norm and an inner product are said to be associated if for all x ∈ X.

* Every quadratic Bézier curve is also a cubic Bézier curve, and more generally, every degree n Bézier curve is also a degree m curve for any m > n. In detail, a degree n curve with control points P < sub > 0 </ sub >, …, P < sub > n </ sub > is equivalent ( including the parametrization ) to the degree n + 1 curve with control points P '< sub > 0 </ sub >, …, P '< sub > n + 1 </ sub >, where.

* Every unital real Banach algebra with no zero divisors, and in which every principal ideal is closed, is isomorphic to the reals, the complexes, or the quaternions.

* Every commutative real unital Noetherian Banach algebra with no zero divisors is isomorphic to the real or complex numbers.

Every cladogram is based on a particular dataset that was analyzed with a particular method.

Every city expansion would imply a new circle ( canals together with town walls ).

Jimmy Hatlo's They'll Do It Every Time was often displayed in a two-panel format with the first panel showing some deceptive, pretentious, unwitting or scheming human behavior and the second panel revealing the truth of the situation.

Every year, the European Joint Conferences on Theory and Practice of Software ( ETAPS ) sponsors the International Conference on Compiler Construction ( CC ), with papers from both the academic and industrial sectors.

Every play is subject to booth review with coaches only having one challenge.

According to Every, one example may be " the myth of St. George " and other stories about saints battling dragons, which were " modelled no doubt in many cases on older representations of the creator and preserver of the world in combat with chaos ".

* Every three years since Earnhardt's death, someone associated with Earnhardt has won the Daytona 500 ( his son Dale Earnhardt, Jr. in 2004, Earnhardt's replacement Cup driver Kevin Harvick in 2007, and Earnhardt Ganassi's Jamie McMurray in 2010.

Every song they wrote was written with an eye toward giving it " deep hidden meaning " or D. H. M.

In Norse mythology, Draupnir ( Old Norse " the dripper ") is a gold ring possessed by the god Odin with the ability to multiply itself: Every ninth night eight new rings ' drip ' from Draupnir, each one of the same size and weight as the original.

Every system must have a class designated as " root ", with one of its creation procedures designated as " root procedure ".

Every year, hundreds of new titles are published in Esperanto along with music.