Problem Solving

300! is divisible by (24!)^n. what is the max. possible integral value of n?

a .13

b.12

c. 15

d.14

e.23

IMO 12


Till n=12,the multiple of 24(24*n)<300

What is OA

greatest prime factor of 24! is 23
so number of 23 in 300! will determine the number of 24! in 300!

this is (300/23)=13

Ans option A

liferocks wrote:greatest prime factor of 24! is 23
so number of 23 in 300! will determine the number of 24! in 300!
this is (300/23)=12
Ans option B

But 23 *13 = 299

liferocks wrote:greatest prime factor of 24! is 23
so number of 23 in 300! will determine the number of 24! in 300!

this is (300/23)=12

Ans option B

There u r !! Correct!

kstv wrote:

liferocks wrote:greatest prime factor of 24! is 23
so number of 23 in 300! will determine the number of 24! in 300!
this is (300/23)=12
Ans option B

But 23 *13 = 299

yup..should be 13..edited.
@gmatmachoman..12 cannot be correct..should be 13..or did i got the process wrong?

source ?

Excuse me, but how do u know that the greatest prime factor in 24! is 23? Thanks

24! can be expanded like 24*23*22*21...........*1

the prime factors in the above list are 2,3,5,7,11,13,17,19,23

So 23 is the greatest prime factor.

so number of 23 in 300! will determine the number of 24! in 300!

Dont really get why we look for the greatest prime factor?

vinay89 wrote:so number of 23 in 300! will determine the number of 24! in 300!

Dont really get why we look for the greatest prime factor?

300 will contain way more of the smaller prime factors. Thus, the number of times the greatest prime factor (of 24) can go into 300 will limit the number of times 24! can go into 300!

For example, if we think about the really small prime factors of 24--the 2s and the 3s, it is clear that there are way more of them in 300. Let's think about the second biggest prime factor of 24--19. Every 19th number is a multiple of 19. But every 23rd number is a multiple of 23. So, there are fewer 23s in 300 than there are 19s. (300/19 = 15.7. Thus, there are 15 19s in 300. This doesn't mean that 24! can divide 300! 15 times, however, because there are only 13 23s in 24!. This means that 24! can't divide 300 a 14th time).

What's the OA?

What if we take, for ease, the case of number of time 6! is divisible by 4!. Should we say since 3! can divide 6! twice, 4! should as well? I think one should be in doubt, if they had one more option like "not feasible to get an integral value of 'n'" or "none" or something like that. This is my first post,correct me if I am wrong.