Answer:
### (9-1) Finding the Minimum Number of Stacks
Radha wants to stack 30 English books and 54 Mathematics books in such a way that each stack has the same number of books of a single subject.
To achieve this, we need to find the greatest common divisor (GCD) of 30 and 54, as this will determine the maximum number of books per stack for each subject such that each stack has the same number of books.
**Step 1: Find the GCD of 30 and 54.**
- The prime factors of 30 are \( 2, 3, \) and \( 5 \) (since \( 30 = 2 \times 3 \times 5 \)).
- The prime factors of 54 are \( 2 \) and \( 3^3 \) (since \( 54 = 2 \times 3^3 \)).
The common prime factors are 2 and 3. The lowest powers of these common factors in 30 and 54 are:
- For 2, the lowest power is \( 2^1 \).
- For 3, the lowest power is \( 3^1 \).
Therefore, the GCD is \( 2 \times 3 = 6 \).
**Step 2: Determine the number of stacks.**
- For English books: \( 30 \div 6 = 5 \) stacks.
- For Mathematics books: \( 54 \div 6 = 9 \) stacks.
So, the minimum number of stacks possible, with each stack containing 6 books of a single subject, is **5 stacks**.
### (9-2) Arranging Science Books
Sona brings 70 Science books and wants to arrange them in the same manner as Radha did for English and Mathematics books, meaning each stack will also contain 6 books.
**Step 1: Determine the number of Science stacks.**
- The number of stacks is \( 70 \div 6 = 11 \) stacks with a remainder.
**Step 2: Calculate the remainder.**
- \( 70 \div 6 = 11 \) stacks with \( 70 - 11 \times 6 = 70 - 66 = 4 \) books left over.
So, there will be **4 Science books** left over after arranging them in stacks of 6.
