What is the number of distinct primes dividing 12! + 13! +14! ?

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Answer 1 · 2016-11-09T16:08:26

#2,3,5,7,11#

#12!+13!+14! =12!(1+13+13 xx 14)#

The primes in #12!# are

#2,3,5,7,11#

and the primes in #(1+13+13 xx 14)# are

#2,7#

so the primes dividing #12!+13!+14! #

are

#2,3,5,7,11#

Answer 2 · 2017-07-11T12:42:29

Five distinct primes divide #12!+13!+14!# and these are #{2,3,5,7,11}#

#12!+13!+14!#

= #12!(1+13+14xx13)#

= #12!(14xx14)#

= #12xx11xx10xx9xx8xx7xx6xx5xx4xx3xx2xx14xx14#

= #ul(2xx2xx3)xx11xxul(2xx5)xxul(3xx3)xxul(2xx2xx2)xx7xxul(2xx3)xx5xxul(2xx2)xx3xx2xxul(2xx7)xxul(2xx7)#

= #2^12xx3^5xx5^2xx7^3xx11#

Hence, five distinct primes divide #12!+13!+14!# and these are #{2,3,5,7,11}#