Opalg said:[sp]In any set of ten consecutive integers, each of the numbers $0,1,\ldots,9$ will occur as the last digit of one of those integers. Select the four integers ending in $1$, $3$, $7$ and $9$. None of those will be divisible by $2$ or $5$. At most two of them will be divisible by $3$ (because consecutive odd multiples of $3$ differ by $6$), and at most one of them will be divisible by $7$ (because consecutive odd multiples of $7$ differ by $14$). So at least one of those four numbers, $N$ say, is not divisible by $3$ or $7$ (or by $2$ or $5$). Apart from $2$, $3$, $5$ and $7$, no other prime can be a factor of more than one integer in a consecutive run of ten. Therefore none of the prime factors of $N$ occurs in any of the other nine numbers, and so $N$ is coprime to all of them.[/sp]
Co-prime numbers, also known as relatively prime numbers, are a pair of numbers that have no common factors except for 1. In other words, their greatest common divisor is 1.
To determine if two numbers are co-prime, you can find the greatest common divisor (GCD) of the two numbers. If the GCD is 1, then the numbers are co-prime.
In a series of 10 numbers, the co-prime numbers are important because they have no common factors with any other number in the series. This means they are relatively "untouched" by other numbers in the series and can provide unique insights or patterns.
In a series of 10 numbers, there can be up to 4 co-prime numbers. For example, in the series 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, the co-prime numbers are 1, 3, 7, and 9.
No, co-prime numbers are not always consecutive in a series of 10. In fact, there could be no co-prime numbers in a series of 10. For example, in the series 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, there are no co-prime numbers.