Inspired by Sal Gard

Computer Science Level 4

200 ! + ( 200 ! ) ! + 3 \large 200! + (200!)! + 3

Find the second smallest prime factor of the number above.

Notation : ! ! denotes the factorial notation. For example, 8 ! = 1 × 2 × 3 × ⋯ × 8 8! = 1\times2\times3\times\cdots\times8 .

Inspiration .


The answer is 158449.

Christopher Boo by Christopher Boo , 24, Singapore

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1 solution

Sam Bealing
Jul 9, 2016

( 200 ! ) ! (200!)! is a very big number so is not easy to handle even by a computer however note that if p < 200 ! ⟹ p ∣ ( 200 ! ) ! p<200! \implies p \vert (200!)! which basically means that any prime factor of 200 ! + 3 200!+3 will divide 200 ! + ( 200 ! ) ! + 3 200!+(200!)!+3 relying on p < 200 ! p<200! (which is still very large). 

1
 
Do[If[Mod[200! + 3, Prime[p]] == 0, Print[Prime[p]]], {p, 1, 100000}]

Prints the first 3 3 prime factors as 3 , 158449 , 238789 3,158449,238789 so, as these are all less than 200 ! 200! the answer is:

158449 \boxed{\boxed{158449}}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

Good recognition that the prime must divide 200 ! + 3 200!+3 , which avoids having to deal with ( 200 ! ) ! (200!)! .

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