Different ways of arranging 72 books in equal piles
just list the factors of 72, and work with that, depending on how many piles you want:
1 72
2 36
2 2 18
2 2 2 9
...
3 3 8
3 24
and so on
Hila zz
To find the different ways of arranging 72 books into equal piles, we need to determine the factors of 72. The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.
Let's go through each factor and calculate the number of piles:
1. If we have 1 pile, all 72 books will be in that pile.
2. If we have 2 piles, each pile will have 36 books.
3. If we have 3 piles, each pile will have 24 books.
4. If we have 4 piles, each pile will have 18 books.
5. If we have 6 piles, each pile will have 12 books.
6. If we have 8 piles, each pile will have 9 books.
7. If we have 9 piles, each pile will have 8 books.
8. If we have 12 piles, each pile will have 6 books.
9. If we have 18 piles, each pile will have 4 books.
10. If we have 24 piles, each pile will have 3 books.
11. If we have 36 piles, each pile will have 2 books.
12. If we have 72 piles, each pile will have 1 book.
So, there are 12 different ways of arranging 72 books into equal piles: 1 pile of 72 books, 2 piles of 36 books, 3 piles of 24 books, 4 piles of 18 books, 6 piles of 12 books, 8 piles of 9 books, 9 piles of 8 books, 12 piles of 6 books, 18 piles of 4 books, 24 piles of 3 books, 36 piles of 2 books, and 72 piles of 1 book.
To find the different ways of arranging 72 books into equal piles, we need to factorize 72 and find its divisors.
Step 1: Factorize 72
The prime factorization of 72 can be written as 2^3 × 3^2.
Step 2: Find the Divisors
To find the divisors of 72, we can use the exponents of each prime factor.
For the factor of 2, we can have powers ranging from 0 to 3:
2^0, 2^1, 2^2, 2^3
For the factor of 3, we can have powers ranging from 0 to 2:
3^0, 3^1, 3^2
Multiplying these combinations together gives us all the divisors of 72:
2^0 × 3^0 = 1
2^1 × 3^0 = 2
2^2 × 3^0 = 4
2^3 × 3^0 = 8
2^0 × 3^1 = 3
2^1 × 3^1 = 6
2^2 × 3^1 = 12
2^3 × 3^1 = 24
2^0 × 3^2 = 9
2^1 × 3^2 = 18
2^2 × 3^2 = 36
2^3 × 3^2 = 72
Step 3: Consider the Arrangement
Since we want to arrange the books into equal piles, the number of piles should be one of the divisors we found in Step 2.
For each divisor, the number of books in each pile would be given by 72 divided by the divisor.
For example:
If we have 1 pile, there would be 72 books in each pile (72 ÷ 1 = 72).
If we have 2 piles, there would be 36 books in each pile (72 ÷ 2 = 36).
And so on for each divisor.
Thus, the different ways of arranging 72 books into equal piles are:
- 1 pile of 72 books
- 2 piles of 36 books each
- 3 piles of 24 books each
- 4 piles of 18 books each
- 6 piles of 12 books each
- 8 piles of 9 books each
- 9 piles of 8 books each
- 12 piles of 6 books each
- 18 piles of 4 books each
- 24 piles of 3 books each
- 36 piles of 2 books each
- 72 piles of 1 book each