What is the minimum value of N that will make this statement true:
If we pick any N composite numbers from 1 to 1000, then we can find 2 numbers whose greatest common divisor is not 1.
It is 72
To find the minimum value of N that will make the given statement true, we need to determine the maximum number of composite numbers we can select from 1 to 1000 without finding any two numbers with a greatest common divisor (GCD) other than 1.
A composite number is a positive integer greater than 1 that can be divided by at least one positive integer other than itself and 1. We can use this definition to identify all composite numbers between 1 and 1000.
To do this, we'll use the Sieve of Eratosthenes algorithm:
1. Create a list of numbers from 2 to 1000.
2. Start with the smallest number, 2, and mark it as prime.
3. Mark all multiples of 2 as composite.
4. Move to the next unmarked number, which is 3, and mark it as prime.
5. Mark all multiples of 3 as composite.
6. Repeat steps 4 and 5 until you reach the square root of 1000.
7. All unmarked numbers that remain are primes.
By applying the Sieve of Eratosthenes, we can identify all the composite numbers between 1 and 1000.
Now, we have the list of composite numbers. To find the minimum value of N, we need to select N composite numbers such that they do not share any common factors between them. This means that we cannot choose any pairs of composite numbers that have a common prime factor.
To maximize the number of composite numbers we can select without finding any two numbers with a GCD other than 1, we need to choose them in such a way that they have distinct prime factors.
To find the minimum value of N, we can start by selecting the first N different prime numbers from the list of primes found. Since each prime number is distinct, the pairs of numbers we form will have a GCD of 1.
For example, the first 6 prime numbers are 2, 3, 5, 7, 11, and 13. If we choose the first 6 composite numbers from the list of primes (4, 6, 8, 9, 10, 12), we can see that no two numbers have a GCD other than 1.
Hence, the minimum value of N that will make the statement true is 6.