If $15\times 14\times 13\times ...\times 3\times 2\times 1=3^m\times n$ where m and n are positive integers, then what is the maximum value of m?

Correct Option (B)

The problem requires determining the maximum exponent 'm' of 3 in the prime factorization of 15!. This is equivalent to finding the total number of times the prime factor 3 appears in the product $1\times 2\times 3\times ...\times 15$.

To find the exponent of a prime 'p' in $n!$, the standard method involves summing the floor values of $n$ divided by successive powers of $p$:

$E_p(n!)=\lfloor \frac{n}{p}\rfloor +\lfloor \frac{n}{p^2}\rfloor +\lfloor \frac{n}{p^3}\rfloor +...$

In this specific case, \(n = 15\) and \(p = 3\). We calculate the terms as follows:

• First term: \(\left\lfloor \frac{15}{3} \right\rfloor = 5\). This counts the multiples of 3 (3, 6, 9, 12, 15), each contributing at least one factor of 3.
• Second term: \(\left\lfloor \frac{15}{3^2} \right\rfloor = \left\lfloor \frac{15}{9} \right\rfloor = 1\). This accounts for multiples of 9 (only 9 within the range 1 to 15), which contribute an additional factor of 3 (since 9 = 3 x 3, one factor was already counted in the first term).
• Third term: \(\left\lfloor \frac{15}{3^3} \right\rfloor = \left\lfloor \frac{15}{27} \right\rfloor = 0\). Higher powers of 3 will also yield 0, as \(3^k > 15\) for \(k \ge 3\).

Summing these values: \(5 + 1 + 0 = 6\).

Therefore, the maximum value of m is 6.

Incorrect Options:

• Option (A) 7: This value is incorrect. It may result from an arithmetic error or a misapplication of the method for counting prime factors in a factorial.
• Option (C) 5: This value is incorrect. It would be obtained if only the first term $\lfloor \frac{15}{3}\rfloor$ was considered, thereby omitting the additional factor of 3 contributed by multiples of 9 (e.g., 9 contributes two factors of 3, but only one would be accounted for in the first term).
• Option (D) 4: This value is incorrect and represents a significant underestimation of the number of prime factors of 3 in 15!.