Finding numbers that satisfy a-2 | a^4+2

stan

New member
Joined
Dec 4, 2018
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6
I need help with finding all numbers that satisfy the following:
\(\displaystyle a-2|a^4+2\)

Should I somehow expand the \(\displaystyle a^4+2\) part or can someone help me get started?
Thanks
 

Subhotosh Khan

Super Moderator
Staff member
Joined
Jun 18, 2007
Messages
18,241
Did you mean:

find all numbers that satisfy the following:
(a−2)|(a^4+2)

If you did - then edit and fix your post. Those parentheses are important.
 

HallsofIvy

Elite Member
Joined
Jan 27, 2012
Messages
4,934
a^4+ 2= (a-2+2)^4+ 2= (a-2)^4+ 4(2)(a-2)^3+ 6(4)(a-2)^2+ 4(8)(a- 2)+ 18= (a- 2)((a-2)^3i+ 8(a-2)^2 24(a- 2)+ 32)+ 18. That will be divisible by a- 2 if and only if 18= 3(3)(2) is divisible by a- 2. And that means that a- 2= 1 so a= 3, or a- 2= 2 so a=4, and so on …

If a= 3, a^4+ 2= 83 which is obviously divisible by a- 2= 1.
If a= 4, a^4+ 2= 258 which is divisible by a- 2= 2.

Continue …
 
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