number-theory-Question-If-a-b-c-N-then-prove-a-b-c-a-b-c-m-n-july-970-

Question Number 112613 by mnjuly1970 last updated on 08/Sep/20

\dots . n u m b e r t h e o r y \dots

Q u e s t i o n : I f a, b, c \in N; t h e n

p r o v e :::

a! * b! * c! ∣ (a + b + c)!

You can't use 'macro parameter character #' in math mode

Commented by Aina Samuel Temidayo last updated on 08/Sep/20

Oh! Sorry, I thought it was an

equation .

Commented by MJS_new last updated on 08/Sep/20

then please give a counterexample

Answered by MJS_new last updated on 08/Sep/20

just a try \dots

let a ⩽ b ⩽ c; obviously c! ∣ (a + b + c)!

\frac{(a + b + c)!}{c!} = (c + 1) (c + 2) \dots (c + a + b)

the number of the factors = a + b

b! = 1 \times 2 \times 3 \times \dots \times b

now each factor of b! is included at least once

in b factors of the form (c + 1) (c + 2) \dots (c + b)

this should be easy to see :

if c = b + n \Rightarrow b ∣ (c + b - n); 1 ⩽ n ⩽ b

the remaining question is, how to show the

“ {remaining}^{″} number is divisible by a

let b + c = n

then \frac{(a + n)!}{n!} = (n + 1) (n + 2) \dots (a + n)

same argument as above

Commented by mnjuly1970 last updated on 09/Sep/20

t h a n k y o u s i r . e x c e l l e n t

a n d a d m i r a b l e . .