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标题: 这题怎么做 [打印本页]

作者: echoetmoi    时间: 2013-5-19 07:19     标题: 这题怎么做

How many powers of 900 are in 50!?这题怎么做啊?
作者: cclovelife    时间: 2013-5-20 06:32

Find the power of 2:
\frac{50}{2}+\frac{50}{4}+\frac{50}{8}+\frac{50}{16}+\frac{50}{32}=25+12+6+3+1=47

= 2^{47}


Find the power of 3:
\frac{50}{3}+\frac{50}{9}+\frac{50}{27}=16+5+1=22

=3^{22}


Find the power of 5:
\frac{50}{5}+\frac{50}{25}=10+2=12

=5^{12}

We need all of them (2,3,5) to be represented twice in 900, 5 can provide us with only 6 pairs, thus there is 900 in the power of 6 in 50!
900^6

To elaborate:

50!=900^xa=(2^2*3^2*5^2)^x*a, where x is the highest possible value of 900 and a is the product of other multiples of 50!.

50!=2^{47}*3^{22}*5^{12}*b=(2^2*3^2*5^2)^6*(2^{35}*3^{10})*b=900^{6}*(2^{35}*3^{10})*b, where b is the product of other multiples of 50!. So x=6.
作者: echoetmoi    时间: 2013-5-21 06:39

I got it.many  thanks !@
作者: songxueb    时间: 2013-5-23 06:37

我是这样考虑的:900=2*2*3*3*5*5
其中5出现的频率最低,所以只需要考虑50!有几个25的幂

1-50中,5的倍数出现10次,其中两次为25的倍数
所以50!中有(10+2)/2=6 个25的幂

所以50!中有6个900的幂...
作者: konglxd    时间: 2013-5-24 06:14

那个只考虑5的方法好厉害~~~
但是真的没人和我一样看成了50!?这种感叹号的吗。。。




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