LinkBack URL
About LinkBacksHi,
I am having difficulty with some proofs.
Show that every composite integer of the form 4n+3 has a factor of the form 4m+3.
and
Show that every composite integer of the form 6n+5 has a factor of the form 6m+5.
[SOLVED] Proof of divisibility
Hi,
I am having difficulty with some proofs.
Show that every composite integer of the form 4n+3 has a factor of the form 4m+3.
and
Show that every composite integer of the form 6n+5 has a factor of the form 6m+5.
I do this one.Originally Posted by R.Laramee
Note the number is not even, so its prime factors are all odd. Hence they have the form. Note that if all prime factors had the form
then their overall product also have the same form. Which is impossible because the number has the form
. This means at least of these prime factors must have the form
Suppose x is composite and of the form 4n+3, and let a and b be such that
ab=x. Then a is of the form 4m, 4m+1 or 4m+2 for some m, and b is of the
form 4k, 4k+1, or 4k+2. But then ab is not of the form 4n+3, a contradiction.
RonL
  |
