* 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 2g ' 2g matrix with < sup >*</ sup > arises as the Seifert matrix of a knot with genus g Seifert surface.

Every processor or processor family has its own machine code instruction set.

Every instruction consists of a 9-bit opcode, a 4-bit register code, and a 23-bit effective address field, which consists in turn of a 1-bit indirect bit, a 4-bit register code, and an 18-bit offset.

Every variable load or constant requires its own separate Load instruction, instead of being bundled within the instruction which uses that value.

This was in part based on clear instruction by Churchill directly to Young, advising him that " Every Part of ( Hong Kong ) Island must be fought over and the enemy resisted with the utmost stubbornness.

Every public school uses Brazilian Portuguese as the medium of instruction, but no law prohibits the use of other languages in private schools.

Every plane that could fly was sent into the air.

Every morning early, in the summer, we searched the trunks of the trees as high as we could reach for the locust shells, carefully detached their hooked claws from the bark where they hung, and stabled them, a weird faery herd, in an angle between the high roots of the tulip tree, where no grass grew in the dense shade.

Every man who dabbles in the market to make a little easy money on the side and suffers losses could at the time hardly face his wife who was wondering how her husband could be so dumb.

Every night they all went to Mama and Papa Albright's, and sat on the open front porch, where they could get the breeze.

Every now and then an atom is replaced by a completely different atom ( and this could be as few as one in a million atoms ).

This was to be based on utilitarian principles ; he said: " Every man has a right to that, the exclusive possession of which being awarded to him, a greater sum of benefit or pleasure will result than could have arisen from its being otherwise appropriated.

Every region in Panama attended the assembly, including the former loyalist region of Veraguas, which was eventually convinced to join the revolution, out of the sheer fact that nothing more could be done for the royalist presence in Panama.

Every person suspected ( owing to his or her immediate attendance on the sick ) could be detained on board for 48 hours or removed to the hospital for a similar period.

The work is much more accessible than Sana ’ i ’ s for instance ; " Every line of the Rubaiyat has more meaning than almost anything you could read in Sufi literature ".

Ebert concludes his review with the following line: " Every time I see the film, I feel a great sadness, that a human imagination could be so limited that it sees its own extinction as a victory.

Every man who could ply an oar pulled up to give Mr Gladstone a cheer.

Every effort was to be made to induce the British to reinforce their forward positions, where the German artillery could engage them.

Every Baronet of Ruddigore since then had fallen under the curse's influence, and died in agony once he could no longer bring himself to continue a life of crime.

Every autumn, according to Plutarch ( Life of Lycurgus, 28, 3 – 7 ), the Spartan ephors would pro forma declare war on the helot population so that any Spartan citizen could kill a helot without fear of blood or guilt ( crypteia ).

Every operating system, even from the same vendor, could have radically different models of commands, operating procedures, and such facilities as debugging aids.

Every person qualified to vote in the district could propose candidates or be nominated as a candidate.

The argument over the quality of Fitzgerald ’ s translation of the Rubaiyat has, according to Dougan, diverted attention from a fuller understanding of the deeply esoteric message contained in Omar ’ s actual material – " Every line of the Rubaiyat has more meaning than almost anything you could read in Sufi literature ".

Every dry scrap of wood they could find, including fence rails, was burned to keep the numerous fires going.

Every ASIC manufacturer could create functional blocks with known electrical characteristics, such as propagation delay, capacitance and inductance, that could also be represented in third-party tools.

Every castle she lived in was equipped with a gymnasium, the Knights ' Hall of the Hofburg was converted into one, mats and balance beams were installed in her bedchamber so that she could practice on them each morning, and the imperial villa at Ischl was fitted with gigantic mirrors so that she could correct every movement and position.

Every hour that Napoleon could have attacked earlier as he did, would have been is his favour, but the French could not attack in the morning for the simple reason that the entire army had not yet taken its battle positions.