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OEF Factoris

OEF Factoris
--- Introduction ---

This module currently contains 15 elementary exercises on
the prime factorization of integers: existence, uniqueness, relation with
gcd and lcm, etc.

Number of divisors

Give an integer
which has exactly positive divisors ( 1 and are divisors of ) and which is divisible by at least
two
three
distinct primes.

Divisors of an integer

Let
be an integer with exactly 3 distinct prime factors:

We know that has divisors more than and that has divisors more than .

Give all possibilities for
. Write one solution
,
,
(separated by a comma) on each line, by increasing order of
.

Division

We have an integer
whose prime factorization is of the form

.

Given that divides
, what is
?

Divisor

We have an integer
whose prime factorization is of the form

.

Given that
divides , what is
?

Sum of factorizations

Let
and
be two positive , having the following factorizations:

,
,

where the factors
are distinct primes.

Is it possible to have a factorization of the form

=

where
are distinct primes?

Find factors II

Here are the prime factorizations of two integers:

,

where the factors
,
are distinct primes. Find these factors.

Find factors III

Here are the prime factorizations of two integers:

where the factors
,
,
are distinct primes. Find these factors.

gcd

Let
and
be two positive integers with the following factorizations:

,

where
,
,
are distinct prime numbers.

Compute
as a function of
,
,
.

lcm

Let
and
be two positive integers with the following factorizations:

,

where
,
,
are distinct prime numbers.

Compute
as a function of
,
,
.

Maximum number of prime factors

Let
be an integer with decimal digits. Given that
has no prime factor < , how many prime factors
may have at most?

Number of divisors II

Let
be a positive integer with the following factorization into distinct prime factors.

What is the number of divisors of
?

A divisor of
is a positive integer which divides
, including 1 and
itself.

Number of divisors III

Let
be a positive integer with the following factorization into distinct prime factors.

What is the number of divisors of
?

A divisor of
is a positive integer which divides
, including 1 and
itself.

Trial division

We have an integer
, and we want to find a prime factor of
by trial dividing
successively by 2,3,4,5,6,... Knowing that
has a prime factorization of the form

where the sum of powers
equal
, but where the factors
are unknown, what is the last divisor we will have to try (without worrying about whether this divisor is prime or not), in the worst case?

Two factors

Compute the number of positive integers
less than or equal to whose prime factorization is of the form

where the powers
and
are integers greater than or equal to .

Two factors II

Compute the number of positive integers
less than or equal to whose prime factorization is of the form

,

where the powers
and
are integers greater than or equal to .
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Description: collection of elementary exercises on the factorization of integers. Serveur Wims de l'ESPE-Nice-Toulon - Université de Nice - Sophia Antipolis

Keywords: interactive mathematics, interactive math, server side interactivity, algebra, arithmetic, number_theory, primes, factorization, integers,factorization, gcd_lcm