Question

Question was provided via image upload.

Explanation:

Identify the mathematical problem

Using the Least Common Multiple and Common Multiples knowledge points

The student's profile indicates a need for conceptual clarification on finding the Least Common Multiple (LCM) of 5 and 6. The blurry image contains a word problem with multiple-choice options.

Let's reconstruct the standard LCM problem for 5 and 6:

• We need to find the smallest positive integer that is a multiple of both 5 and 6.
• Multiples of 5: \(5, 10, 15, 20, 25, 30, 35, \dots\)
• Multiples of 6: \(6, 12, 18, 24, 30, 36, \dots\)
• The smallest common value in both lists is 30.

The options in the blurry image describe scenarios about grouping or finding a common number:

• Option A mentions "each package of... and each package of... contains...".
• Option B mentions "each package of... and each package of... contains...".
• Option C and D discuss whether it is possible to buy the same number of items when they come in packages of 5 and 6.

Since 5 and 6 are relatively prime, their least common multiple is:

$$ \text{LCM}(5, 6) = 5 \times 6 = 30 $$

Thus, to get the same number of items, one must buy a multiple of 30 items (e.g., 30, 60, 90, etc.). The smallest non-zero common number of items is 30.

List multiples to find the LCM

Using the Common Multiples knowledge point

$$
LATEXBLOCK0
$$

Determine the correct option

Using the Least Common Multiple knowledge point

• The smallest common multiple is 30.
• This means the minimum number of items to have an equal amount of both is 30.
• Therefore, the correct choice is the one stating that it is possible to buy the same number of items, and the least number of items is 30.

Answer:

• (A) It is possible to buy the same number of items; the least number of items is 30. (Correct answer)
• (B) It is possible to buy the same number of items; the least number of items is 60.
• (C) It is not possible to buy the same number of items because 5 and 6 do not share common factors.
• (D) It is not possible to buy the same number of items because 5 is odd and 6 is even.