MHB Co-prime Numbers in a Series of 10

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In any set of ten consecutive integers, at least one number is co-prime to the other nine. This is demonstrated by selecting integers ending in 1, 3, 7, and 9, which are not divisible by 2 or 5. Among these, at most two can be divisible by 3 and only one by 7, ensuring that at least one number, denoted as N, is not divisible by 2, 3, 5, or 7. Since no other prime can factor more than one integer in this range, N shares no prime factors with the other nine numbers. Thus, N is co-prime to all the other integers in the set.
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Show that in 10 consecutive numbers there is at least one number which is co-prime to other 9 numbers.
 
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[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]
 
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]

above answer is good and better than my answer which is as below
Out of 10 consecutive numbers 5 numbers are divisible by 2 and not more that 4 numbers are divisible by 3 out of which
maximum 2 numbers are odd and divisible by by 3 that makes 7, 2 numbers are divisible by 5 out of which is even so there are maximum one number is divisible by 5 and neither 2 nor 3 and that makes 8 and maximum 2 numbers are divisible by 7 out of which is only one is odd maximum one number is divisible by 7 and neither 2 nor 3 nor 5 and that makes maximum 9 numbers that are divisible by one of 2,3,5,7. so there is at least one number which is not divisible by 2,3,5 or 7 so the lowest prime factor of the same is 11 and it cannot divide any other number of the set. so this number is co-prime to rest 9.
 
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