Grade 12Algebra
What is the Common factor of (13^7+11^7) and (13^5+11^5)
What is the Common factor of (13^7+11^7) and (13^5+11^5)
1 Answer
Arun
For n odd, we have the algebraic identities
x^n + y^n = (x + y)(x^(n-1) - x^(n-2)y + x^(n-3)y^2 + ... + x^2 y^(n-3) - xy^(n-2) + y^(n-1)).
Therefore,
13^7 + 11^7 = (13 + 11)(13^6 - 13^5*11 + 13^4*11^2 - 13^3*11^3 + 13^2*11^4 - 13*11^5 + 11^6)
13^5 + 11^5 = (13 + 11)(13^4 - 13^3*11 + 13^2*11^2 - 13*11^3 + 11^4)
and it follows that 24 is a common factor of (13^7 + 11^7) and (13^5 + 11^5).
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