perplexus.info :: Just Math : Power GCD

Power GCD (Posted on 2025-05-09)

For any natural number a, show that gcd(a³+1, a⁷+1)=a+1.

No Solution Yet Submitted by Danish Ahmed Khan

No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)

I think this does it too

Comment 2 of 2 |

Show gcd(a^3+1, a^7+1)=a+1.

Clearly a+1 does divide both expressions:

a^3+1 = (a+1) * (a^2 - a + 1)

a^7+1 = (a+1) * (a^6 - a^5 + a^4 - a^3 + a^2 - a + 1)

But for it to be the greatest, the other factors must be relatively prime.

Let Numerator = (a^6 - a^5 + a^4 - a^3 + a^2 - a + 1)

Let Denominator = (a^2 - a + 1)

Numerator / Denominator = (a^4 - a) with remainder 1/(a^2 - a + 1)

Posted by Larry on 2025-05-09 12:25:53

Please log in:

Forums (1)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (6)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:


perplexus dot info