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

Nov 9, 2016

$2 , 3 , 5 , 7 , 11$

#### Explanation:

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

The primes in 12! are

$2 , 3 , 5 , 7 , 11$

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

$2 , 7$

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

are

$2 , 3 , 5 , 7 , 11$

Jul 11, 2017

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

#### Explanation:

12!+13!+14!

= 12!(1+13+14xx13)

= 12!(14xx14)

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

= $\underline{2 \times 2 \times 3} \times 11 \times \underline{2 \times 5} \times \underline{3 \times 3} \times \underline{2 \times 2 \times 2} \times 7 \times \underline{2 \times 3} \times 5 \times \underline{2 \times 2} \times 3 \times 2 \times \underline{2 \times 7} \times \underline{2 \times 7}$

= ${2}^{12} \times {3}^{5} \times {5}^{2} \times {7}^{3} \times 11$

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