Question

If I multiply all whole numbers from 1 through 100, the largest power of 4 that is a factor of the product is
A: 4^25
B: 4^32
C: 4^48
D: 4^50

Answer 1

Where do you get these questions? -.-

Answer 2

Lol. From a contest I took this year.

Answer 3

Sadly, I failed the contest. Contest too hard. T~T

Answer 4

hint: list all the multiples of 4 from 1 to 100

Answer 5

LIST all them multiples of from 1 to 100?
There's 25 number that is a multiple of 4.

Answer 6

LOL Okay, annoying...

Answer 7

I'm quite certain of my answer, but explaining it... gah...

Answer 8

Say we have the product from 1 to 100 of all integers...

Answer 9

you'll have 25 multiples of 4

you'll also have 50 multiples of 2....they pair up (in any way you want) to have 25 pairs which are multiples of 2*2 = 4

so you really have 25+25 = 50 copies of 4 being multiplied

Answer 10

Yep.

Answer 11

In fact, let's have it in prime factors...

Answer 12

and now, let's take away all the 2's
Actually, we need to count how many times 2 is multiplied in the factorisation of the product of all integers from 1 to 100.

Answer 13

@jim_thompson5910 are you accounting for the fact that half the multiples of 2 are already accounted for as multiples of 4? And that a factor of 2 is only 4^(1/2)? So shouldn't we have:
25 multiples of 4
50 multiples of 2 - 25 multiples of 4 already counted = 25 new multiples of 2 = 12.5 multiples of 4?

Answer 14

oh true

Answer 15

If I absolutely had to guess I would say C. I'm honestly not sure how I would solve this problem.

Answer 16

so (multiples of 2) + (multiples of 4) - (multiples of 2 and 4) = (25)+(25) - (25) = 25

Answer 17

So... there are 50 multiples of 2, so that gives us 50 2's.
There are 25 multiples of 4, so that gives us a FURTHER 25 2's
There are 12 multiples of 8, so that gives us a FURTHER 12 2's
There are 6 multiples of 16, so that gives us a FURTHER 6 2's
There are 3 multiples of 32, so that gives us a FURTHER 3 2's
There is 1 multiple of 64, so that gives us a FURTHER 1 2.
Add those up, you get
50+25+12+6+3+2 = 96 2's

So \[\Large 2^{96}=4^{48}\]is your man :D

Answer 18

I think he nailed it

Answer 19

Wowow. You too good TJ.

Answer 20

It happens :D

Answer 21

bow to TJ* I want to be your student. :3

Answer 22

LOL that'll be the day :D

Answer 23

:D