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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