To find the greatest number of bags possible while ensuring each bag has the same number of balloons and stickers, we need to determine the greatest common divisor (GCD) of the two quantities: 100 balloons and 68 stickers.
First, we calculate the GCD of 100 and 68.
- The prime factorization of 100 is \( 2^2 \times 5^2 \).
- The prime factorization of 68 is \( 2^2 \times 17 \).
The common factors are \( 2^2 \), which means the GCD is \( 4 \).
Now, we can determine how many balloons and stickers can go into each bag:
- For balloons: \( \frac{100}{4} = 25 \)
- For stickers: \( \frac{68}{4} = 17 \)
Thus, the maximum number of bags you can make is 4, and each bag can contain 25 balloons and 17 stickers.
So the answer is 25 balloons and 17 stickers.