Лозан Михаилов Иванов, ф. №43851
ФМИ, Информатика, гр.3, 1курс
Курсова работа по английски език, 2003/2004
Introduction in Number Theory
http://www.math.com/school/numbertheory/lesson1.html
Definition of Division. For every two positive integers a and b, we say that a divides b if
and only if there exists a positive integer c with b=a.c. We will notify it by a/b.
Now we prove some properties of the division relation.
1.For every positive integer a we have a/a. This is obvious since a=a.1
2.If a/b then a<b or a=b. Because there exists a positive integer c with b=a.c and since c
is an integer, the property is obvious.
3.If a/b and a/c, then a/b+c(and a/b-c if b>c). We have b=k.a and c=l.a where k and l are
positive integers and therefore b+c=a.(k+l) so by definition we have a/b+c.
4.If a/b and c is a positive integer, then a/bc. Since b=a.k we have b.c=a(kc) so a/bc.
The Division Theorem.
For every thwo positive integers a and b exists unique non-negative integers q and r with
the property
a=b.q+r, r<b.
(q is called “quotient” and r is called “remainder”)
Proof.
1.Uniqueness. Suppose there are two pairs of non-negative integers (q,r) and (s,t) with
a=b.q+r, r<b and a=b.s+t, t<b.
By substracting the above equations we get
b|q-s|=|t-r|
wich shows that b/|t-r| and since |t-r|<b, we have |t-r|=0 and therefore t=r which implies
that |q-s|=0 so q=s. Therefore (q,r)=(s,t) and there is at most one pair with the required
properties.
2.Existence. If b/a there is a positive iteger q with a=b.q, now set r=0 and we are done. If
b does not divide a, since k.b is unbound when k gets large and 0<a, there is such
positive integer k with (k-1).b<a<k.b. Denote q=k-1 and r=a-qb. Obviously, a=b.q+r,
and 0<a-(k-1).b<b, so 0<r<b.
The Theorem is now proved.
The Greatest Common Divider(GCD) and the Least Common Multiple(LCM).
Definition.If d/a and d/b we say that d is a common divider of a and b. Since 1/a and
1/b, and d does not exceed min(a,b), there is a finite number of common dividers for
every two positive integers a and b. Therefore, there is a common divider greater than all
other common dividers, and we will denote it by the greatest common divider of a and b,
or by GCD(a,b).
Definition.If a/d and b/d we say that d is a common multiple of a and b. Since d is
greater than or equal to each of a and b, the set of all common multiples of a and b is
limited from bellow, so it has a smallest element which we will denote by the least
common multiple of a and b, or by LCM(a,b).
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Лозан Михаилов Иванов, ф. №43851
Glossary
Uses the glossary of http://www.math.com/school/glossary/glossindex.html
integers
The set of numbers containing zero, the natural numbers, and all the negatives of
the natural numbers.
division
The process of calculating a / b = c.
quotient
The integer answer q to the division problem a/b=c, where a=b.q+r.
remainder
The integer r, where a=b.q+r.
greatest common divider (GCD)
The largest number that divides two or more numbers.
least common multiple (LCM)
The smallest positive integer that is a multiple of two or more numbers.
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Лозан Михаилов Иванов, ф. №43851
Упражнение
A relation R is called “Ordering” if:
1)aRa;
2)if aRb and bRc, then aRc;
3)if aRb and bRa, then a=b.
Show that the ralation “/” is ordering, in oder words, you have to show that to show that:
1)a/a;
2)if a/b and b/c, then a/c;
3)if a/b and b/a, then a=b.
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Лозан Михаилов Иванов, ф. №43851
Решение на Упражнението
1)This is the first property of division and is proved on page 1.
2)Since a/b and b/c we have b=ka and c=lb for some k and l. Therefore c=(kl)a,
which means that a/c.
3)Since a/b and b/a we have b=ka and a=lb for some k and l. Therefore a=(kl)a,
so kl=1. On the other hand, k and l are positive integers which implies that k=l=1,
so a=b.
NB: 3) obvious follows from the second property of division, because a/b and b/a
implie that a is not greater than b and that b is not greater than a, so a=b.
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