Question

The given equation $x^3+y^4=7$ has;

• A. no solution in integers
• B. one solution in integers
• C. two solution in integers
• D. three solution in integers

Answer 1

Answer: (A)

no solution in integers

The residues of a cube module $13$ are $0,1,5,8$ or $12$ while the residues of a fourth power are $0,1,3$ or $9$ module $13$. This leads us to conclude that the equation $x^3+y^4\equiv 7$ (mod $13$) cannot hold.