To solve this problem, we need to find the greatest number of boxes that will have an equal number of math books and an equal number of science books in each box. This means we are looking for the greatest common divisor (GCD) of 63 (math books) and 42 (science books).

Step-by-step solution:

1. Prime factorization of 63 and 42:

   • $63 = 3^2 \times 7$
   • $42 = 2 \times 3 \times 7$

2. The GCD is found by taking the product of the lowest powers of the common prime factors:

   • The common prime factors are $3$ and $7$.
   • The lowest powers of these are $3^1$ and $7^1$.

   Therefore, the GCD is: $GCD(63, 42) = 3 \times 7 = 21$

Conclusion:

The greatest number of boxes that can be used is 21. Each box will contain:

• $\frac{63}{21} = 3$ math books, and
• $\frac{42}{21} = 2$ science books.

Would you like more details or have any further questions?

Related Questions:

1. What is the least common multiple (LCM) of 63 and 42?
2. How would you distribute 72 math books and 56 science books equally in boxes?
3. Can the number of boxes change if one additional science book is added?
4. What if there were 84 math books and 63 science books?
5. How many ways can you divide 63 math books and 42 science books with unequal numbers per box?

Tip: The greatest common divisor (GCD) is useful for determining the maximum way to equally divide items into groups or boxes.