Divisibility of the factorial of 50
Task number: 3848
What is the highest power of 5 that divides \(50!\)? Determine the general formula for a prime number \( p \) and the factorial of a number \(n\).
Solution
For each multiple of 5 in the set \( \{1{,}2,…, 50 \} \) 1 is added to the exponent over 5 in the prime decomposition of \( 50! \).
Similarly, for each multiple of \( 25 = 5 ^ 2 \), an additional 1 is added.
Specifically, multiples of \( 5 \) in the decomposition \( 50! \) occur \( \left \lfloor \frac {50} 5 \right \rfloor \)-times and multiples of \( 25 \) apperar \( \left \lfloor \frac{50}{25} \right \rfloor \)-times. In total, 5 is contained in the prime decomposition of \( 50! \) twelve times.
The general formula for the primes \( p_1 = 2, p_2 = 3,… \) is
\(n!=p_1^{k_1}p_2^{k_2}\cdots p_r^{k_r}\), kde \( k_i=\sum\limits_{j=1}^{\lfloor\log_{p_i}n\rfloor} \left\lfloor\frac{n}{p_i^j}\right\rfloor\)
Answer
The highest power of five that divides the expression \( 50! \) is the twelfth power of five.
