Final answer:
To prove that given any 17 integers, there exist nine of them whose sum is divisible by 9, we can use the fact that given any five integers, we can find three whose sum is divisible by 3.
Step-by-step explanation:
To prove that given any 17 integers, there exist nine of them whose sum is divisible by 9, we can use the fact that given any five integers, we can find three whose sum is divisible by 3.
We can start by selecting any five integers from the given 17.
Since there are 17 integers and we need to find nine whose sum is divisible by 9, we can choose any five integers from the given list. Let's say we choose the first five integers: 3, 4, 5, 7, and 7.
We know that the sum of any three integers whose sum is divisible by 3 will also be divisible by 3.
So, among the five integers we chose, there must be three integers whose sum is divisible by 3.
Let's say those three integers are 3, 4, and 7.
Now, we can repeat this process by selecting any five integers from the remaining 12 integers.
Since there are enough integers left, we can continue this process until we have selected nine integers whose sum is divisible by 9.