Science:Math Exam Resources/Courses/MATH312/December 2012/Question 02 (c)/Solution 1

The answer is true.

Recall that 7 passes Miller's test if

$7^d\equiv 1mod25$

or the following holds for some r

$7^{2^{r}d}\equiv -1mod25$

where $25-1=24=2^{s}d=2^3\cdot 3$ and $0\le r\le 3-1$.

If 7 passes the test, then 25 is a probable prime. If it fails the test, then the number is not prime.

So we check manually:

$\begin{array}{rl}{7}^3& \equiv 49\cdot 7\equiv -1\cdot 7\equiv ̸1mod25\\ 7^{2^1\cdot 3}& \equiv 49^3\equiv (-1{)}^3\equiv -1mod25\end{array}$

Thus, 7 passes Miller's test. (Note that 25 is not prime but still 7 passes Miller's test).