Website owner: James Miller
Divisors, factors and multiples of integers; common divisor, greatest common divisor, common multiple, least common multiple, division algorithm, Euclid’s algorithm, unique factorization theorem
a divides b. We say an integer a divides an integer b (written "a | b") if there exists an integer c such that b = ac. When a | b we say that a is a factor of b, a is a divisor of b, or b is a multiple of a.
Example. 2 | 10 since 10 = 2·5 . The integer 10 is a multiple of 2.
Note the following: 3 divides 0 and, in general, a | 0 for all a ε I . Why? It follows as a direct consequence of the above definition since 0 = a·0 . This is a case of a result that would not be expected from the general concept of b as being a multiple of a, but which follows as a consequence of the axiomatic style definition.
Theorem 1. If a | b and a | c then a | (bx + cy) for all integers x,y.
Prime number. An integer p which is not 0 or +1 and is divisible by no integers except +1 and +p.
Common divisor of two or more quantities. A quantity which is a factor of each of the quantities. A common divisor of 10, 15, and 75 is 5; a common divisor of x2 - y2 and x2 - 2xy + y2 is x - y , since x2 - y2 = (x - y)(x + y) and x2 - 2xy + y2 = (x - y)2.
Syn. common factor, greatest common measure.
Greatest common divisor (g.c.d.) of two or more quantities. A common divisor that is divisible by all other common divisors. For positive integers, the greatest common divisor is the largest of all common divisors e.g. the common divisors of 30 and 42 are 2, 3, and 6, the largest being the greatest common divisor 6. If we consider negative integers the common divisors of 24 and 60 are 1, 2, 3, 4, 6, and 12. and the greatest common divisors are 12. Syn. greatest common factor, greatest common measure.
The greatest common divisor of a and b is denoted by (a,b).
Common multiple of two or more quantities. A quantity which is a multiple of each of two or more given quantities. The number 6 is a common multiple of 2 and 3. x2 - 1 is a common multiple of x - 1 and x + 1.
Least common multiple (l.c.m.) of two or more quantities. The least quantity that is exactly divisible by each of the given quantities; 12 is the l.c.m. of 2, 3, 4, and 6. The l.c.m. of a set of algebraic quantities is the product of all their distinct prime factors, each taken the greatest number of times it occurs in any one of the quantities; the l.c.m. of x2 - 1 and x2 - 2x + 1 is (x - 1)2(x + 1) .
Tech. The l.c.m. of a set of quantities is a common multiple of the quantities which divides every common multiple of them.
Division Algorithm. For any integer a and any positive integer b, there exist unique integers q and r such that
a = bq + r, 0 r < b
The integer a is the dividend, b is the divisor, q is the quotient and r is the remainder. [Note. This theorem could be stated differently as “the quotient a/b equals q plus a remainder of r” – which explains the terminology.] For polynomials, the division algorithm states that, for any polynomial f and any non-constant polynomial g, there exist unique polynomials q and r such
where either r = 0 or the degree of r is less than the degree of g. The polynomials f, g, q, and r are the dividend, divisor, quotient, and remainder.
Euclid’s algorithm. A method of finding the greatest common divisor (g.c.d.) of two numbers – one number is divided by the other, then the second by the remainder, the first remainder by the second remainder, the second by the third, etc. When exact division is finally reached, the last divisor is the greatest common divisor of the given numbers (integers). In algebra, the same process can be applied to polynomials. E.g., to find the greatest common divisor of 12 and 20, we have 20 12 is 1 with remainder 8; 12 8 is 1 with remainder 4; and 8 4 = 2; hence 4 is the g.c.d.
Theorem 2. Any two integers a ≠ 0 and b ≠ 0 have a positive greatest common divisor (g.c.d.) which can be expressed as a “linear combination” of a and b in the form d = au + bv for integers u and v.
Theorem 3. If p is a prime, then p | ab implies p | a or p | b.
Theorem 4. If p is a prime and if p is a divisor of the product a∙b∙c∙ .... ∙t of n integers, then p is a divisor of at least one of these integers.
Relatively prime integers. Two integers a and b are said to be relatively prime if
(a,b) = 1.
The unique factorization Theorem. Every integer a > 1 has a unique factorization, except for order, into a product of primes
1. James/James. Mathematics Dictionary.
The Way of Truth and Life
God's message to the world
Jesus Christ and His Teachings
Words of Wisdom
Way of enlightenment, wisdom, and understanding
Way of true Christianity
America, a corrupt, depraved, shameless country
On integrity and the lack of it
The test of a person's Christianity is what he is
Who will go to heaven?
The superior person
On faith and works
Ninety five percent of the problems that most people have come from personal foolishness
Liberalism, socialism and the modern welfare state
The desire to harm, a motivation for conduct
The teaching is:
On modern intellectualism
On Self-sufficient Country Living, Homesteading
Principles for Living Life
Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.
America has lost her way
The really big sins
Theory on the Formation of Character
You are what you eat
People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest
Cause of Character Traits --- According to Aristotle
These things go together
We are what we eat --- living under the discipline of a diet
Avoiding problems and trouble in life
Role of habit in formation of character
The True Christian
What is true Christianity?
Personal attributes of the true Christian
What determines a person's character?
Love of God and love of virtue are closely united
Walking a solitary road
Intellectual disparities among people and the power in good habits
Tools of Satan. Tactics and Tricks used by the Devil.
On responding to wrongs
Real Christian Faith
The Natural Way -- The Unnatural Way
Wisdom, Reason and Virtue are closely related
Knowledge is one thing, wisdom is another
My views on Christianity in America
The most important thing in life is understanding
Sizing up people
We are all examples --- for good or for bad
Television --- spiritual poison
The Prime Mover that decides "What We Are"
Where do our outlooks, attitudes and values come from?
Sin is serious business. The punishment for it is real. Hell is real.
Self-imposed discipline and regimentation
Achieving happiness in life --- a matter of the right strategies
Self-control, self-restraint, self-discipline basic to so much in life
We are our habits
What creates moral character?