Hi.(adsbygoogle = window.adsbygoogle || []).push({});

I have a question.

I know the rule of the hightest power of 2 dividing n!, the highest power means if [tex]2^{k}|n![/tex], k is the greatest value. For example,

n the hightest power of 2 dividing n!

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1 0

2 1

3 1

4 3

5 3

6 4

7 4

8 7

9 7

10 8

11 8

12 10

13 10

14 11

15 11

16 15

17 15

18 16

19 16

20 18

21 18

22 19

23 19

24 22

25 22

26 23

27 23

28 25

29 25

30 26

31 26

32 31

33 31

Let f(n) is the highest power of 2 dividing n!, I think if [tex]2^{k}=n[/tex], then [tex]f(n) = 2^{k}-1[/tex], else let [tex]n=2^{k_{1}}+2^{k_{2}}+...++2^{k_{r}}[/tex], then [tex]f(n)=f(2^{k_{1}})+f(2^{k_{2}})+...+f(2^{k_{r}})=2^{k_{1}}-1+2^{k_{2}}-1+...+2^{k_{r}}-1[/tex], more verification of n shows my rule is correct, but I can't prove that.

Help me to prove my rule. thanks.

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# Find the highest power of 2 dividing n!

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