Discussion
Find the highest power of 12 that divides 50!.
Answer: A
To find the highest power of 12, we need to find the highest powers of its prime factors, which are \(2^2\) and 3.
Power of 3 in 50! = \(\lfloor 50/3 \rfloor + \lfloor 50/9 \rfloor + \lfloor 50/27 \rfloor = 16+5+1 = 22\).
Power of 2 in 50! = \(\lfloor 50/2 \rfloor + \lfloor 50/4 \rfloor + \lfloor 50/8 \rfloor + \lfloor 50/16 \rfloor + \lfloor 50/32 \rfloor = 25+12+6+3+1 = 47\).
Since \(12=2^2 \times 3\), we need pairs of (one 3) and (two 2s). We have 22 threes. For the twos, we have 47, so we can make \(\lfloor 47/2 \rfloor = 23\) pairs of 2s. The power of 12 is limited by the smaller of these counts (22 for threes, 23 for pairs of twos). The limiting factor is the number of 3s. So the highest power of 12 is 22.
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