Coprime
In mathematics, the integers a and b are said to be coprime or relatively prime iff they have no common factor other than 1 and -1, or equivalently, if their greatest common divisor is 1.For example, 6 and 35 are coprime, but 6 and 27 are not because they are both divisible by 3. 1 is coprime to every integer; 0 is coprime only to 1 and -1.
A fast way to determine whether two numbers are coprime is given by the Euclidean algorithm.
Properties
The numbers a and b are coprime if and only if there exist integers x and y such that ax + by = 1 (see Bézout's identity). Equivalently, b has a multiplicative inverse modulo a: there exists an integer y such that by ≡ 1 (mod a).
If a and b are coprime and a divides a product bc, then a divides c.
If a and b are coprime and bx ≡ by (mod a), then x ≡ y (mod a). In other words: b yields a unit in the ring Za of integers modulo a.
The two integers a and b are coprime if and only if the point with coordinates (a,b) in an Cartesian coordinate system is "visible" from the origin (0,0), in the sense that there is no point with integer coordinates between the origin and (a,b).
The probability that two randomly chosen integers are relatively prime is 6/π2 (see Pi).
Two natural numbers a and b are coprime if and only if the numbers 2a-1 and 2b-1 are coprime.
Generalization
Two ideals A and B in the commutative ring R are called coprime if A + B = R. This generalizes Bezout's identity. If A and B are coprime, then AB = A∩B; furthermore, if C is a third ideal such that A contains BC, then A contains C.
With this definition, two principal ideals (a) and (b) in the ring of integers Z are coprime if and only if a and b are coprime.
See also: Greatest common divisor
I shall stay in you, friend, for some few hours.
And climb your throat on sobs, until it's chased
To do without what blood remained me from my wound.
Disabled
He sat in a wheeled chair, waiting for dark,
Legless, sewn short at elbow. Through the park
Voices of play and pleasure after day,
When glow-lamps budded in the light-blue trees
-- In the old times, before he threw away his knees.
Girls' waists are, or how warm their subtle hands,
For it was younger than his youth, last year.
He's lost his colour very far from here,
And half his lifetime lapsed in the hot race,
One time he liked a bloodsmear down his leg,
It was after football, when he'd drunk a peg,
Someone had said he'd look a god in kilts.
That's why; and maybe, too, to please his Meg,
He asked to join. He didn't have to beg;
Germans he scarcely thought of; and no fears
For daggers in plaid socks; of smart salutes;
Esprit de corps; and hints for young recruits.
Only a solemn man who brought him fruits
Now, he will spend a few sick years in Institutes,
And take whatever pity they may dole.
Passed from him to the strong men that were whole.
And put him into bed? Why don't they come?
The flourish of loud clouds, the Chariot throne,
And from the bronze west long retreat is blown,
Shall Life renew these bodies? Of a truth
Or fill these void veins full again with youth
"My head hangs weighed with snow."
My fiery heart sinks aching. It is death.
Nor my titanic tears the seas be dried."
[End of original text.]
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