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What is the largest non-negative integer #k# such that #24^k# divides 13! ?
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1 Answer
Nov 9, 2016

Answer:

#k = 3#

Explanation:

Using the properties of exponents that #(ab)^x = a^xb^x# and #(a^x)^y = a^(xy)#, we have

#24^k = (2^3*3^1)^k = (2^3)^k*(3^1)^k = 2^(3k)*3^k#

Thus #13!# is divisible by #24^k# if and only if #13!# is divisible by #2^(3k)# and is divisible by #3^k#.

We can tell the greatest power of #2# by which #13!# is divisible by if we look at its factors which are divisible by #2#:

#2 = 2^1#
#4 = 2^2#
#6 = 2^1*3#
#8 = 2^3#
#10 = 2^1*5#
#12 = 2^2*3#

As none of the odd factors contribute any factors of #2#, we have

#13! = (2^1*2^2*2^1*2^3*2^1*2^2)*m = 2^(10)*m#

where #m# is some integer not divisible by #2#. As such, we know that #13!# is divisible by #2^(3k)# if and only if #2^10# is divisible by #2^(3k)#, meaning #3k <= 10#. As #k# is an integer, this means #k <= 3#.

Next, we can look at which factors of #13!# are divisible by #3#:

#3 = 3^1#
#6 = 3^1 * 2#
#9 = 3^2#
#12 = 3^1*4#

As no other factors of #13!# contribute any factors of #3#, this means

#13! = (3^1*3^1*3^2*3^1)*n = 3^5*n#

where #n# is some integer not divisible by #3#. As such, we know that #3^5# is divisible by #3^k#, meaning #k <= 5#.

The largest nonnegative integer satisfying the constraints #k<=3# and #k<=5# is #3#, giving us our answer of #k=3#.

A calculator will verify that #(13!)/24^3 = 450450#, whereas #(13!)/24^4=18768.75#