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. interactive exercises, online calculators and plotters, mathematical recreation and games
- Keywords: interactive mathematics, interactive math, server side interactivity, algebra, arithmetic, number_theory, primes, factorization, integers,factorization, gcd_lcm