The problem asks for the greatest number of plates that Lola can make using all 63 spring rolls and 84 cheese cubes, while ensuring that each plate has the same number of both spring rolls and cheese cubes.

To solve this, we need to determine the greatest common divisor (GCD) of 63 and 84, as this will give us the greatest number of plates that Lola can make with the same number of spring rolls and cheese cubes on each plate.

Step 1: Find the GCD of 63 and 84

• Prime factorization of 63:
  $63 = 3^2 \times 7$
• Prime factorization of 84:
  $84 = 2^2 \times 3 \times 7$

The common factors are $3 \times 7 = 21$.

So, the GCD of 63 and 84 is 21.

Step 2: Calculate the number of each appetizer per plate

• Number of spring rolls per plate:
  $63 \div 21 = 3$
• Number of cheese cubes per plate:
  $84 \div 21 = 4$

Final Answer:

• The greatest number of plates Lola can make is 21.
• Each plate will have 3 spring rolls and 4 cheese cubes.

Would you like further details or have any questions?

Here are five related questions:

1. How would the solution change if Lola had 72 spring rolls and 90 cheese cubes?
2. What if Lola had only 42 spring rolls and 70 cheese cubes—how many plates could she make?
3. How can the least common multiple (LCM) be applied in problems like this one?
4. How does knowing prime factorization help in finding the GCD quickly?
5. What strategies can you use if the numbers are much larger, say 630 and 840?

Tip: When faced with problems involving equal division, always consider using the GCD to maximize the number of groups!