how many factors of 60 are greater than 根号60？

Eileenl · 发表于 2010-11-17 22:52:26

如题。

sdcar2010 · 发表于 2010-11-17 23:08:13

60, 30, 20, 15, 12, 10

Eileenl · 发表于 2010-11-18 16:50:16

要一个一个suan?omg

bydgoodandevil · 发表于 2010-11-18 16:58:17

先看60一共有多少个因子
60=2^2*3*5
所以一共有(2+1)(1+1)(1+1)=12个因子
大于根号60的因子就是12/2=6个

Eileenl · 发表于 2010-11-18 17:10:25

（2+1).....那个怎么来的？

bydgoodandevil · 发表于 2010-11-18 18:14:50

就是那几个质因子的指数加1再相乘

Eileenl · 发表于 2010-11-18 19:48:31

就是那几个质因子的指数加1再相乘

-- by 会员 bydgoodandevil (2010/11/18 18:14:50)

3Q~

fudanwyh · 发表于 2010-11-19 05:24:37

就是那几个质因子的指数加1再相乘

-- by 会员 bydgoodandevil (2010/11/18 18:14:50)

why so?

sdcar2010 · 发表于 2010-11-19 06:13:10

Suppose a number X = (2^m) * (3^n) * (5^p), wherein m, n, and p are positive integers.

For a number Y to be a factor of X, Y needs to be expressed as Y = (2^a) * (3^b) * (5^c), wherein a can be chosen among 0, 1, . . . m; b can be chosen among 0, 1, . . . n; and c can be chosen among 0, 1, . . . p. Therefore, the number of possible combinations of m,n,p for Y is : (m+1) * (n+1) * (p+1). The number of possible factors for X is (m+1) * (n+1) * (p+1)

Then among all these factors, at most only one factor when times itself equals X. The rest are in pairs, the product of which equals X. For the paired factors, one is greater than X^(1/2), while the other is smaller than X^(1/2).

Therefore, let's calculate (m+1) * (n+1) * (p+1). If the product is an even number Z, then Z/2 would be the number of factors that will be greater than X^(1/2). If the product is an odd number Z, then (Z - 1)/2 would be the number of factors that will be greater than X^(1/2).

For example, 15 = 3 * 5 has 4 factors, two of which are greater than 15^(1/2); while 16 = 2^4 has 5 factors (including 4, the square root of 16!), two of which are greater than 16^(1/2).

how many factors of 60 are greater than 根号60？

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