#
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: = _{1}_{2}_{3} , = _{1}_{2}_{4} , where the factors _{i} are *distinct* primes.

Is it possible to have a factorization of the form

| | = _{1}_{2}_{3} , 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 = _{1}^{1} _{2}^{2} ... _{t}^{t} 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:
Factorization
Integers
arithmetics

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- Description: collection of elementary exercises on the factorization of integers. This is the main site of WIMS (WWW Interactive Multipurpose Server): interactive exercises, online calculators and plotters, mathematical recreation and games
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