Factorial Calculation and Prime Powers

The expression \(1 \times 2 \times 3 \times 4 \times \dots \times 30\) represents the factorial of 30, denoted as \(30!\). We need to find the largest power of 3 that divides \(30!\). This is equivalent to finding the exponent of the prime number 3 in the prime factorization of \(30!\).

Prime Factorization of Factorials

To find the exponent of a prime number \(p\) in the prime factorization of \(n!\), we can use Legendre's formula. The formula states that the exponent \(E_p(n!)\) is the sum of the quotients obtained by dividing \(n\) by successive powers of \(p\).

Legendre's Formula:

\(E_p(n!) = \sum_{k=1}^{\infty} \left\lfloor \frac{n}{p^k} \right\rfloor = \left\lfloor \frac{n}{p} \right\rfloor + \left\lfloor \frac{n}{p^2} \right\rfloor + \left\lfloor \frac{n}{p^3} \right\rfloor + \dots\)

In this problem, \(n = 30\) and the prime number is \(p = 3\). We need to calculate \(E_3(30!)\).

Calculating the Exponent of 3 in 30!

We apply Legendre's formula by dividing 30 by successive powers of 3:

• First term: Divide 30 by \(3^1 = 3\). The floor of the result is \(\left\lfloor \frac{30}{3} \right\rfloor = \left\lfloor 10 \right\rfloor = 10\). This counts the numbers from 1 to 30 that are multiples of 3 (3, 6, 9, ..., 30).
• Second term: Divide 30 by \(3^2 = 9\). The floor of the result is \(\left\lfloor \frac{30}{9} \right\rfloor = \left\lfloor 3.33\dots \right\rfloor = 3\). This counts the numbers from 1 to 30 that are multiples of 9 (9, 18, 27). These numbers contribute an additional factor of 3 besides the one already counted as multiples of 3.
• Third term: Divide 30 by \(3^3 = 27\). The floor of the result is \(\left\lfloor \frac{30}{27} \right\rfloor = \left\lfloor 1.11\dots \right\rfloor = 1\). This counts the numbers from 1 to 30 that are multiples of 27 (only 27). This number contributes a third factor of 3.
• Fourth term: Divide 30 by \(3^4 = 81\). The floor of the result is \(\left\lfloor \frac{30}{81} \right\rfloor = \left\lfloor 0.37\dots \right\rfloor = 0\). Since the power of 3 is now greater than 30, all subsequent terms will be 0.

Now, we sum the results of the floor divisions:

\(E_3(30!) = 10 + 3 + 1 + 0 + \dots = 14\)

The exponent of 3 in the prime factorization of \(30!\) is 14. This means that the largest power of 3 that divides \(30!\) is \(3^{14}\).

Largest Power of 3

The calculation shows that the highest power of 3 that is a factor of \(1 \times 2 \times \dots \times 30\) is \(3^{14}\).