Točno
Oct. 16, 2015, 4:39 p.m. (10 years, 11 months)
Let
,
,
,
,
,
be positive integers and let
.
Suppose that the number
divides
and
. Prove that
is composite.
%V0
Let $a$, $b$, $c$, $d$, $e$, $f$ be positive integers and let $S = a+b+c+d+e+f$.
Suppose that the number $S$ divides $abc+def$ and $ab+bc+ca-de-ef-df$. Prove that $S$ is composite.
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Pretpostavimo suprotno,
je prost.
Neka je
Iz pretpostavki zadatka vrijedi za svaki
cijeli broj:
Sto je apsurdno jer
%V0
Pretpostavimo suprotno, $S$ je prost.
Neka je $P(x) = (x+a)(x+b)(x+c), Q(x) = (x+d)(x+e)(x+f)$
Iz pretpostavki zadatka vrijedi za svaki $x$ cijeli broj: $ S \mid P(x) + Q(-x)$
$x = d \implies S \mid P(d) = (d+a)(d+b)(d+c) \underset{BSOMP}{\implies} S \mid d+a $
Sto je apsurdno jer $0 < d+a < S$
| Oct. 28, 2015, 1:42 p.m. | grga | Točno |
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