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OEF factoris
OEF factoris
--- Introduction ---
This module actually contains 14 elementary exercises on
the prime factorization of integers: existence, uniqueness, relation with
gcd and lcm, etc.
Number of divisors
Give an integer
which have exactly divisors ( 1 and are divisors of ) and which is divisible by at least
two
three
distinct primes.
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:
= 123 , = 124 ,
where the factors i are distinct primes.
Is it possible to have a factorization of the form
| | = 123 ,
where i are distincts 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 m, n be two positive integers with the following factorizations.
m = , n = ,
where , , are distinct prime numbers.
Compute gcd(m,n) as a function of , , .
lcm
Let m, n be two positive integers with the following factorizations.
m = , n = ,
where , , are distinct prime numbers.
Compute lcm(m,n) as a function of , , .
Maximum of factors
Let be an integer with decimal digits. Given that has no prime factor < , how many prime factors may have at maximum?
Number of divisors II
Let be a positive integer with the following factorization into distinct prime factors.
= 12
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.
= 123
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
= 1122 ... tt
where the sum of powers 1+2+...+t = , (but where the factors i 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 whose prime factorization is of the form
= × ,
where the powers and are integers .
Two factors II
Compute the number of positive integers whose prime factorization is of the form
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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, prime, prime factorization, integer, factor, gcd,lcm
.
whose prime factorization is of the form