To determine the maximum value of
k in the expression
26!=nâ‹…8k, we need to consider the prime factorization of 8 and the number of times each of its prime factors appear in 26 !.
First, notice that 8 can be factored into primes as follows:
8=23Therefore, for
8k to be a factor of 26 !, it should be equivalent to
23k. To find the maximum value of
k, we need to determine the highest power of 2 that divides 26 !.
The power of a prime
p in
n ! is given by the sum:
⌊‌⌋For our problem, we need to calculate the highest power of 2 in 26 !. This can be computed as follows:
⌊‌⌋+⌊‌⌋+⌊‌⌋+⌊‌⌋Evaluating each term, we get:
⌊‌⌋=⌊13⌋=13⌊‌⌋=⌊6.5⌋=6⌊‌⌋=⌊3.25⌋=3⌊‌⌋=⌊1.625⌋=1Adding these together, we find:
13+6+3+1=23 Therefore, the highest power of 2 that divides 26 ! is
223.
Since
8k=23k, the value of
k must satisfy:
3k≤23Simplifying, we have:
k≤⌊‌⌋=7Thus, the maximum value of
k is 7 .