What is the highest power of 2 that will divide 90! ?

From 1 to 90 there are 90 numbers, so 90! has 90 factors which we can rearrange to our pleasing

Because you said that a power a 2 must divide 90! we'll only be interested in even numbers.

We'll factor out all the 2's we can find until there are no more even numbers.

90! = 1*2*3*4*5*6*7*8*9*...*85*86*87*88*89*90

90! = (2*4*6*8*10*...*84*86*88*90)*(1*3*5*7*9*...*85*87*89)

90! = (2^45)*(1*2*3*4*15*...*42*43*44*45)*(1*3*5*7*9*...*85*87*89)

90! = (2^45)*(2*4*6*8*10*...*42*44)*(1*3*5*7*9*...*41*43*45)*(1*3*5*7*9*...*85*87*89)

90! = (2^45)*(2^22)*(1*2*3*4*...*21*22)*(1*3*5*7*9*...*41*43*45)*(1*3*5*7*9*...*85*87*89)

90! = (2^45)*(2^22)*(2*4*6*8*...*20*22)*(1*3*5*7*...*19*21)*(1*3*5*7*9*...*41*43*45)* (1*3*5*7*9*...*85*87*89)

90! = (2^45)*(2^22)*(2^11)*(1*2*3*4*...*10*11)*(1*3*5*7*...*19*21)*(1*3*5*7*9*...*41*43*45)* (1*3*5*7*9*...*85*87*89)

90! = (2^45)*(2^22)*(2^11)*(2*4*6*8*10)*(1*3*5*7*9*11)*(1*3*5*7*...*19*21)*(1*3*5*7*9*...*41*43*45)* (1*3*5*7*9*...*85*87*89)

90! = (2^45)*(2^22)*(2^11)*(2^5)*(1*2*3*4*5)*(1*3*5*7*9*11)*(1*3*5*7*...*19*21)*(1*3*5*7*9*...*41*43*45)* (1*3*5*7*9*...*85*87*89)

90! = (2^45)*(2^22)*(2^11)*(2^5)*2*(2^2)*(1*3*5)*(1*3*5*7*9*11)*(1*3*5*7*...*19*21)*(1*3*5*7*9*...*41*43*45)* (1*3*5*7*9*...*85*87*89)

90! = [2^(45+22+11+5+1+2)]*(1*3*5)*(1*3*5*7*9*11)*(1*3*5*7*...*19*21)*(1*3*5*7*9*...*41*43)* (1*3*5*7*9*...*85*87*89)

90! = [2^86]*(1*3*5)*(1*3*5*7*9*11)*(1*3*5*7*...*19*21)*(1*3*5*7*9*...*41*43)* (1*3*5*7*9*...*85*87*89)

90! = (2^86) * [odd_number]

Your answer is 86 is the highest power of 2 that divides 90!

Best regards!

The highest power of 2 that will divide 90! is 2^86.

I just counted all the"2" factors of the numbers from 1 to 90.

45 + 22 + 11 + 5 + 2 + 1 = 86

45 even, + 22 for the multiples of 4, plus 11 for the multiples of 8, and so on...

2^1.

how i got the answer -

i thought about it.

2^1 will.

2^2 won't.

2^3 won't.

2^4 won't...

obviously nothing else is going to. anyway, 90/2^1 is an odd number.

If we let {x} be floor of x, it is {90/2}+{90/4}+{90/8}...=86.

For proof, please ask me on my website below.