1: Find all n, 1<=n<=7770, n divisible by 7, n not div. by k if 7 doesn't divide k &

1: Find all n, 1<=n<=7770, n divisible by 7, n not div. by k if 7 doesn't divide k &

Help in combinatorics exercise
Hey everyone I'm stuck in the exercise, I will be happy to work out a solution

Find the number of integers n, where 1 < n < 7770, which have the following attributes:

. . .a. n is divisible by 7

. . .b. for any natural number k such that \(\displaystyle k\, \neq\, 7\, \leq\, k\, \leq\, 10\),
. . . . .it is true that n is not divisible by k

Thanks friends.

yossa said:

Help in combinatorics exercise
Hey everyone I'm stuck in the exercise, I will be happy to work out a solution

Find the number of integers n, where 1 < n < 7770, which have the following attributes:

. . .a. n is divisible by 7

. . .b. for any natural number k such that \(\displaystyle k\, \neq\, 7\, \leq\, k\, \leq\, 10\),
. . . . .it is true that n is not divisible by k

Thanks friends.

Click to expand...

I would start by thinking about what the question means. You are counting numbers from 1 through 7770 that are divisible by 7, but not by 2, 3, 4, 5, 6, 8, 9, or 10. You can simplify that by focusing on prime divisors, since for example a number divisible by 10 is already divisible by 2 and 5. So the question reduces to

How many numbers from 1 through 7770 are divisible by 7, but not by 2, 3, or 5?

What methods can you think of for this counting? Perhaps the inclusion-exclusion principle might be useful.