Determine whether the integer 701 is prime by testing?

[Math100]

Homework Statement

Determine whether the integer $701$ is prime by testing all primes $p\leq\sqrt{701}$ as possible divisors. Do the same for the integer $1009$.

Relevant Equations

None.

Proof:

Consider all primes $p\leq\sqrt{701}\leq 27$.
Note that $701=2(350)+1$
$=3(233)+2$
$=5(140)+1$
$=7(100)+1$
$=11(63)+8$
$=13(53)+12$
$=17(41)+4$
$=19(36)+17$
$=23(30)+11$.
Thus, no prime numbers less than $27$ are divisible by the integer $701$.
Therefore, the integer $701$ is prime.
Now, we consider all primes $p\leq\sqrt{1009}\leq 32$.
Note that $1009=2(504)+1$
$=3(336)+1$
$=5(201)+4$
$=7(144)+1$
$=11(91)+8$
$=13(77)+8$
$=17(59)+6$
$=19(53)+2$
$=23(43)+20$
$=29(34)+23$
$=31(32)+17$.
Thus, no prime numbers less than $32$ are divisible by the integer $1009$.
Therefore, the integer $1009$ is prime.

 

[Office_Shredder]

I didn't check all the numbers but it looks like you're doing the right thing