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Solved on Dec 07, 2023

Find the largest value of $n$ such that $5^{n}$ divides $1 \times 2 \times 3 \times \cdots \times 15$ .

STEP 1

Assumptions
1. We are looking for the highest power of $5$ , denoted as $5^n$ , that divides the product $1 \times 2 \times 3 \times \ldots \times 15$ without leaving a remainder.
2. We are to consider only the numbers from $1$ to $15$ inclusive.

STEP 2

Identify all multiples of $5$ in the given range $1$ to $15$ .
$Multiples\, of\, 5: 5, 10, 15$

STEP 3

Determine the power of $5$ in each multiple of $5$ .
- $5$ is $5^1$ . - $10$ is $2 \times 5^1$ . - $15$ is $3 \times 5^1$ .

STEP 4

Count the total number of $5$ s present in the factors.
- There is one $5$ in $5$ . - There is one $5$ in $10$ . - There is one $5$ in $15$ .

STEP 5

Add the powers of $5$ from each multiple of $5$ .
$Total\, powers\, of\, 5 = 1 (from\, 5) + 1 (from\, 10) + 1 (from\, 15)$

STEP 6

Calculate the sum of the powers of $5$ .
$Total\, powers\, of\, 5 = 1 + 1 + 1 = 3$

STEP 7

Conclude that the largest value of $n$ such that $5^n$ is a factor of the product $1 \times 2 \times \ldots \times 15$ is $3$ .
The largest value of $n$ is $3$ .

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