What is the highest power of 5 that divides `90xx80xx70xx60xx50xx40xx30xx20xx10`

Options

A. 10
B. 12
C. 14
D. 15

Answer

To find the highest power of 5 that divides the product \( 90 \times 80 \times 70 \times 60 \times 50 \times 40 \times 30 \times 20 \times 10 \), we will follow these steps:

### Step 1: List the numbers
The numbers we are multiplying are:
- 90
- 80
- 70
- 60
- 50
- 40
- 30
- 20
- 10

### Step 2: Factor each number into its prime factors
We will factor each number to find how many times 5 appears in their prime factorization.

- \( 90 = 2 \times 3^2 \times 5 \) (1 factor of 5)
- \( 80 = 2^4 \times 5 \) (1 factor of 5)
- \( 70 = 2 \times 5 \times 7 \) (1 factor of 5)
- \( 60 = 2^2 \times 3 \times 5 \) (1 factor of 5)
- \( 50 = 2 \times 5^2 \) (2 factors of 5)
- \( 40 = 2^3 \times 5 \) (1 factor of 5)
- \( 30 = 2 \times 3 \times 5 \) (1 factor of 5)
- \( 20 = 2^2 \times 5 \) (1 factor of 5)
- \( 10 = 2 \times 5 \) (1 factor of 5)

### Step 3: Count the total number of factors of 5
Now, we sum the number of factors of 5 from each of the factorizations:
- From \( 90 \): 1
- From \( 80 \): 1
- From \( 70 \): 1
- From \( 60 \): 1
- From \( 50 \): 2
- From \( 40 \): 1
- From \( 30 \): 1
- From \( 20 \): 1
- From \( 10 \): 1

Adding these up:
\[ 1 + 1 + 1 + 1 + 2 + 1 + 1 + 1 + 1 = 11 \]

### Conclusion
The highest power of 5 that divides \( 90 \times 80 \times 70 \times 60 \times 50 \times 40 \times 30 \times 20 \times 10 \) is \( 5^{11} \).

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