## amistre64 3 years ago Let n=\(2^3*3^2*5*7^3*11\) Find the smallest factorial: k! that is divisible by n Show your work!! :)

1. Malachy

27!

2. amistre64

thats not work, thats an answer .... teaches me nothing :/

3. hba

I thought this was open study.

4. Malachy

=2^3*3^13*5*7 here 13 is the maximum power raised on any number. thus we have to take till the number untill 3 comes 13 times 3=1 6=1 9=2 12=1 15=1 18=2 21=2 24=2 27=3 Total of rhs=15>13 thus 27! will be the ans

5. amistre64

------------------ 2 2 2 3 3 5 7 7 7 11 2 ------------------ 1 2 2 3 3 5 7 7 7 11 2 3 ------------------ 2 2 3 5 7 7 7 11 2 3 4 ------------------ 3 5 7 7 7 11 2 3 4 5 ------------------ 3 7 7 7 11 2 3 4 5 6 ------------------ 7 7 7 11 2 3 4 5 6 7 ------------------ 7 7 11 2 3 4 5 6 7 8 9 10 11 ------------------ 7 7 2 3 4 5 6 7 8 9 10 11 12 13 14 ---------------------------- 7 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 ----------------------------------------------- i think its 21!

6. amistre64

you seem to have used something different than that i posted in the question

7. amistre64

does your method, using my posted information, equate to 27 as well? or is it 21?

8. asnaseer

yup - I also get 21 using the same method as @amistre64

9. amistre64

im pretty adept at brute mathing; i see some potential in Malachys if i can get it verified ;)

10. asnaseer

I have never heard of Malachys - oh well - another new topic to learn...

11. asnaseer

do you have a reference (a link) to this method?

12. amistre64

i see in mine the 7^3 produces the biggest parts; and 7*3 = 21

13. amistre64

my method? i thought it up

14. asnaseer

true - but still have to show that all numbers before this are factored out as well - which you did using brute force :)

15. asnaseer

you said " i see some potential in Malachys" <<<--- is this made up? :)

16. amistre64

lol, we may never know unless they show up again ;)

17. asnaseer

you jest with us mere mortals sir! :)

18. amistre64

=2^3*3^2*5*7^3*11 here 3 is the maximum power raised on any number. thus we have to take till the number untill (2 or 7 ???) comes 3 times ???? Total of rhs=??? thus 21! will be the ans seems simple enough to me

19. Malachy

Actually i was missed as 2^3*3^2*5*7*3^11 thats why that ans came

20. amistre64

can you give a generalized rule for your method?