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 ${2,3,5,7,11}$