Questions: Liam is buying bottles of soda in packages that contain 8 bottles each. If the total number of sodas Liam bought was between 45 and 50, how many did he buy? Explain your answer.

Solution Steps

To determine how many bottles of soda Liam bought, we need to find a multiple of 8 that falls within the range of 45 to 50. We can do this by iterating through the range and checking for multiples of 8.

We are given that the total number of sodas Liam bought is between 45 and 50. Therefore, we define the range as: \[ 45 \leq x \leq 50 \]

Next, we need to find the multiples of 8 within this range. The multiples of 8 can be expressed as: \[ x = 8n \quad \text{for integers } n \] We will check which values of \( x \) in the range \( [45, 50] \) satisfy this condition.

We check the integers in the range:

• For \( x = 45 \): \( 45 \mod 8 \neq 0 \)
• For \( x = 46 \): \( 46 \mod 8 \neq 0 \)
• For \( x = 47 \): \( 47 \mod 8 \neq 0 \)
• For \( x = 48 \): \( 48 \mod 8 = 0 \)
• For \( x = 49 \): \( 49 \mod 8 \neq 0 \)
• For \( x = 50 \): \( 50 \mod 8 \neq 0 \)

The only value that satisfies the condition is \( x = 48 \).

Final Answer