Answer
(a) not a prime number.
(b) prime number.
(c) not a prime number.
(d) prime number.
Work Step by Step
(a) As $\sqrt{667}\approx25.8$, we should test divisibility of $667$ by the following prime numbers: $2,3,5,7,11,13,17,19,23$.
Use some known techniques for $2,3,5,11$, we can rule out them all.
Continue to test other numbers, we can find $23$ is a factor, thus $667$ is not a prime number.
(b) As $\sqrt{557}\approx23.6$, we should test divisibility of $557$ by the following prime numbers: $2,3,5,7,11,13,17,19,23$.
Use some known techniques for $2,3,5,11$, we can rule out them all.
Continue to test other numbers, we can not find a factor, thus $557$ is a prime number.
(c) As $\sqrt{527}\approx22.96$, we should test divisibility of $527$ by the following prime numbers: $2,3,5,7,11,13,17,19$.
Use some known techniques for $2,3,5,11$, we can rule out them all.
Continue to test other numbers, we can find a factor $17$, thus $527$ is not a prime number.
(d) As $\sqrt{613}\approx24.8$, we should test divisibility of $613$ by the following prime numbers: $2,3,5,7,11,13,17,19,23$.
Use some known techniques for $2,3,5,11$, we can rule out them all.
Continue to test other numbers, we can not find a factor, thus $613$ is a prime number.
