To find the smallest number of pumpkin seeds Liz could have bought, we start by noting that the pumpkin seeds come in packages of 9 and the sunflower seeds come in packages of 11. Liz bought the same number of each type of seed, so we will denote the number of packages of each seed type she bought as \( x \).
The total number of seeds she bought can be expressed as:
- For pumpkin seeds: \( 9x \)
- For sunflower seeds: \( 11y \)
Since Liz bought the same number of each type of seed, we need to find \( x \) and \( y \) such that: \[ 9x = 11y \] This means we need to find integer values for \( x \) and \( y \). To proceed, we can rearrange this equation: \[ \frac{9x}{11} = y \]
From this equation, it implies that \( 9x \) must be divisible by 11 for \( y \) to be an integer. Thus, we want to find the smallest integer \( x \) such that \( 9x \) is a multiple of 11. This can be done by finding the least common multiple (LCM) of the coefficients 9 and 11.
The formula for the least common multiple \( \text{lcm}(a, b) \) is given by: \[ \text{lcm}(a, b) = \frac{a \cdot b}{\text{gcd}(a, b)} \] Since 9 and 11 are coprime (i.e., their greatest common divisor, \( \text{gcd}(9, 11) \), is 1), the LCM can be calculated as follows: \[ \text{lcm}(9, 11) = \frac{9 \cdot 11}{1} = 99 \]
Thus, \( 9x \) must be equal to 99: \[ 9x = 99 \] From this, we solve for \( x \): \[ x = \frac{99}{9} = 11 \]
To find the corresponding number of pumpkin seeds she bought: \[ \text{Number of pumpkin seeds} = 9x = 9 \cdot 11 = 99 \]
Now we check the sunflower seeds. With \( x = 11 \), we can compute \( y \): \[ 11y = 99 \implies y = \frac{99}{11} = 9 \]
Thus, the number of sunflower seeds she bought is: \[ \text{Number of sunflower seeds} = 11y = 11 \cdot 9 = 99 \]
Hence, Liz bought the same number of seeds for both types: 99 seeds each.
In conclusion, the smallest number of pumpkin seeds Liz must have bought is: \[ \boxed{99} \]
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