# 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.

= 1 2

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.

= 1 2 3

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

= 11 22 ... 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

= × ,

where the powers and are integers .

Other exercises on:
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