Prove that 26! is divisible by 2^10.What is the largest nat. no. n so that 26! is divisible by 2^n?

Answer 1

You should write out the factors of each number to see if it's divisible:

26! = 26*25*24*23*22*21*20......*1
2^10 = 2*2*2*2*2*2*2*2*2*2

All the even numbers of 26! have a factor of 2. Since there are 13 even numbers in 26!, there are at least 13 factors of 2. Since 2^10 only has 10 factors of 2, then it can be divided into 26!.

Using the same logic, the largest natural number 2^n can only be divided into 26! if it has the same number of "2-factors". You need to find the number of 2 factors in each even number of 26!

26 = 2*13
24 = 2*2*2*3
22 = 2*11
20 = 2*2*5
18 = 2*3*3
16 = 2*2*2*2
14 = 2*7
12 = 2*2*3
10 = 2*5
8 = 2*2*2
6 = 2*3
4 = 2*2
2 = 2*1

Counting the number of 2's, you get 23. Therefore the largest 2^n number that can be divisible into 26! is 2^23.

Hope this helps

Answer 2

26!
26 * 25 * 23 * 22 * 21 * 20 * 19 * 18 * 17 * 16 * 15 * 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1

26 = 2 * 13
25 = 5 * 5
24 = 2 * 2 * 2 * 3
23 = 23
22 = 2 * 11
21 = 3 * 7
20 = 2 * 2 * 5
19 = 19
18 = 2 * 3 * 3
17 = 17
16 = 2 * 2 * 2 * 2
15 = 3 * 5
14 = 2 * 7
13 = 13
12 = 2 * 2 * 3
11 = 11
10 = 2 * 5
9 = 3 * 3
8 = 2 * 2 * 2
7 = 7
6 = 2 * 3
5 = 5
4 = 2 * 2
3 = 3
2 = 2
1 = 1

26! = 1 * 2^23 * 3^10 * 5^6 * 7^3 * 11^2 * 13^2 * 17 * 19 * 23

or

(2^10 * 2^13 * 3^10 * 5^6 * 7^3 * 11^2 * 13^2 * 17 * 19 * 23)/2^10

2^13 * 3^10 * 5^6 * 7^3 * 11^2 * 13^2 * 17 * 19 * 23

the highest number that "n" can be in 2^n and 26! is still divisible is "n" = 23

Answer 3

26!=26*25*...
26=2*13
24=2*2*2*3
22=2*11
20=2*2*5
18=2*9
16=2*2*2*2
14=2*7
12=2*2*3
10=2*5
8=2*2*2
6=2*3
4=2*2
2=2
in other words 26! has 23 factors of 2
so 26! is divisible by 2^{23}
and hence also divisible by 2^{10}

Answer 4

[26/2] = 13
[26/4] = 6
[26/8] = 3
[26/16] = 1
n = 13 + 6 + 3 + 1 = 23
23 > 10

Answer 5

Since every other factor of a factorial is even, there are at least 13 twos in the factorial. Powers of 4 have 2 twos, Powers of 8 have 3 twos, powers of 16 have 4 twos.