To solve the problem of determining the minimum number of integers that need to be drawn from the set of integers from 1 to 34 to ensure that at least one pair of these numbers has a product that is divisible by 3, we can follow these steps:
1. Understand the Divisibility by 3:
- A product of two numbers is divisible by 3 if at least one of the numbers is divisible by 3.
- Therefore, to ensure that we have a pair of numbers whose product is divisible by 3, we need at least one number in our set that is divisible by 3.
2. Identify Numbers Divisible by 3 in the Range:
- We need to determine how many numbers between 1 and 34 are divisible by 3. These numbers form the sequence: 3, 6, 9, ..., 33.
- This sequence is an arithmetic series where the first term [tex]\( a_1 = 3 \)[/tex] and the common difference [tex]\( d = 3 \)[/tex].
- We can find the number of terms in this sequence by solving for [tex]\( n \)[/tex] in the equation [tex]\( 3 + (n-1) \times 3 \leq 34 \)[/tex]:
[tex]\[
3n \leq 34 \implies n \leq \frac{34}{3} \approx 11.33
\][/tex]
So, [tex]\( n = 11 \)[/tex] (since [tex]\( n \)[/tex] must be an integer).
3. Identify Numbers Not Divisible by 3:
- There are 34 integers in total. Out of these, 11 are divisible by 3.
- Therefore, there are [tex]\( 34 - 11 = 23 \)[/tex] numbers that are not divisible by 3.
4. Determine the Worst-case Scenario:
- If we were to pick only numbers that are not divisible by 3, how many numbers could we pick without guaranteeing that any product of two of them is divisible by 3?
- In the worst-case scenario, we pick all 23 numbers that are not divisible by 3 before picking any number that is divisible by 3.
5. Ensuring a Pair with a Product Divisible by 3:
- To ensure at least one number whose product with any other number is divisible by 3, we must draw at least one more number after picking all 23 that are not divisible by 3.
- Therefore, the minimum number of integers we need to draw is [tex]\( 23 + 1 = 24 \)[/tex].
Answer:
The minimum number of integers that need to be drawn to ensure that there are two numbers whose product is divisible by 3 is [tex]\( \boxed{23} \)[/tex].
