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

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

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Answer 1 · 2016-11-09T16:08:26

$2, 3, 5, 7, 11$ $2,3,5,7,11$

Explanation:

$12! + 13! + 14! = 12! (1 + 13 + 13 \times 14)$ $12!+13!+14! =12!(1+13+13 xx 14)$

The primes in $12!$ $12!$ are

$2, 3, 5, 7, 11$ $2,3,5,7,11$

and the primes in $(1 + 13 + 13 \times 14)$ $(1+13+13 xx 14)$ are

$2, 7$ $2,7$

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

are

$2, 3, 5, 7, 11$ $2,3,5,7,11$

Answer 2 · 2017-07-11T12:42:29

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

Explanation:

$12! + 13! + 14!$ $12!+13!+14!$

= $12! (1 + 13 + 14 \times 13)$ $12!(1+13+14xx13)$

= $12! (14 \times 14)$ $12!(14xx14)$

= $12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 14 \times 14$ $12xx11xx10xx9xx8xx7xx6xx5xx4xx3xx2xx14xx14$

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

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

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

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

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