Number of divisors
Give an integer
which have exactly $val11 divisors ( 1 and $val11 are divisors of $val11) and which is divisible by at least
two
three
distinct primes.
Division
We have an integer $val8 whose prime factorization is of the form $val8 = $val6$val9×$val7$val10×$val14 . Given that $val18 divides $val8, what is $val8?
Divisor
We have an integer $val8 whose prime factorization is of the form $val8 = $val6$val16$val7$val17 . Given that $val8 divides $val15, what is $val8?
Sum of factorizations
Let $val8 and $val9 be two positive $val14, having the following factorizations: $val8 = $val101$val102$val103 , $val9 = $val101$val102$val104 , where the factors $val10i are distinct primes.
Is it possible to have a factorization of the form
|$val8 $val15 $val9| = $val111$val112$val113 , where $val11i are distincts primes?
Find factors II
Here are the prime factorizations of two integers: $val26 = $val8$val20 $val9$val24 , $val27 = $val8$val21 $val9$val25 , where the factors $val8, $val9 are distinct primes. Find these factors.
Find factors III
Here are the prime factorizations of two integers: $val28 = $val8$val20 $val9$val24 $val10$val27 , $val29 = $val8$val21 $val9$val25 $val10$val27 , where the factors $val8, $val9, $val10 are distinct primes. Find these factors.
gcd
Let m, n be two positive integers with the following factorizations. m = $val15 $val16 $val17 , n = $val18 $val19 $val20 , where $val6, $val7, $val8 are distinct prime numbers.
Compute gcd(m,n) as a function of $val6, $val7, $val8.
lcm
Let m, n be two positive integers with the following factorizations. m = $val15 $val16 $val17 , n = $val18 $val19 $val20 , where $val6, $val7, $val8 are distinct prime numbers.
Compute lcm(m,n) as a function of $val6, $val7, $val8.
Maximum of factors
Let $val6 be an integer with $val8 decimal digits. Given that $val6 has no prime factor < $val9, how many prime factors $val6 may have at maximum?
Number of divisors II
Let $val6 be a positive integer with the following factorization into distinct prime factors. $val6 = $val71$val8 $val72$val9 What is the number of divisors of $val6 ? (A divisor of $val6 is a positive integer which divides $val6, including 1 and $val6 itself.)
Number of divisors III
Let $val6 be a positive integer with the following factorization into distinct prime factors. $val6 = $val71$val8 $val72$val9 $val73$val10 What is the number of divisors of $val6 ? (A divisor of $val6 is a positive integer which divides $val6, including 1 and $val6 itself.)
Trial division
We have an integer $val6 < $val9, and we want to find a prime factor of $val6 by trial dividing $val6 successively by 2,3,4,5,6,... Knowing that $val6 has a prime factorization of the form $val6 = $val71$val81 $val72$val82 ... $val7t$val8t where the sum of powers $val81+$val82+...+$val8t = $val10, (but where the factors $val7i 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 $val6
$val13 whose prime factorization is of the form $val6 = $val10$val8×$val11$val9 , where the powers $val8 and $val9 are integers
$val14.
Two factors II
Compute the number of positive integers $val6
$val13 whose prime factorization is of the form $val6 = $val10$val8×$val11$val9 , where the powers $val8 and $val9 are integers
$val14.