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The problem asks to determine the largest value of \( k \) such that \( 2^k \) is a factor of the product \( 1 \times 2 \times 3 \times \cdots \times 30 \), which is the factorial \( 30! \).

To find the largest value of \( k \) for which \( 2^k \) divides \( 30! \), we need to count how many factors of 2 are in \( 30! \). This can be done by using Legendre's formula, which states that the exponent of a prime \( p \) in \( n! \) is given by:

\[
k = \left\lfloor \frac{n}{p} \right\rfloor + \left\lfloor \frac{n}{p^2} \right\rfloor + \left\lfloor \frac{n}{p^3} \right\rfloor + \cdots
\]

Here, \( n = 30 \) and \( p = 2 \). Let's calculate \( k \):

\[
k = \left\lfloor \frac{30}{2} \right\rfloor + \left\lfloor \frac{30}{4} \right\rfloor + \left\lfloor \frac{30}{8} \right\rfloor + \left\lfloor \frac{30}{16} \right\rfloor + \left\lfloor \frac{30}{32} \right\rfloor
\]

Now, evaluating each term:

\[
\left\lfloor \frac{30}{2} \right\rfloor = 15
\]
\[
\left\lfloor \frac{30}{4} \right\rfloor = 7
\]
\[
\left\lfloor \frac{30}{8} \right\rfloor = 3
\]
\[
\left\lfloor \frac{30}{16} \right\rfloor = 1
\]
\[
\left\lfloor \frac{30}{32} \right\rfloor = 0
\]

Thus, the total \( k \) is:

\[
k = 15 + 7 + 3 + 1 = 26
\]

So, the largest value of \( k \) such that \( 2^k \) is a factor of \( 30! \) is **26**.

Would you like a detailed breakdown or further clarification?

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Here are five related questions for practice:
1. What is the largest \( k \) such that \( 2^k \) divides \( 50! \)?
2. How would you find the largest \( k \) such that \( 3^k \) divides \( 30! \)?
3. Can you calculate the number of factors of 5 in \( 30! \)?
4. What is the sum of the factors of 2 in the factorial of any number \( n! \)?
5. How would this approach change if the prime \( p \) was 7 instead of 2?

**Tip:** Legendre's formula is a powerful tool for finding the highest powers of any prime that divide a factorial.

Consider the product: 1 × 2 × 3 × 4 × ... × 30. Determine the largest value of k such that 2^k is a factor of this product.

Explore how to determine the largest value of k such that 2^k divides 30! using Legendre's formula. This problem covers number theory concepts, including prime factorization and factorial calculations. Suitable for high school students in Grades 9-12.

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