91Ó°ÊÓ

Find the four incongruent solutions of each of the quadratic congruences below: (a) \(x^{2}=15(\bmod 77)\). (b) \(x^{2}=100(\bmod 209)\). (c) \(x^{2} \equiv 58(\bmod 69)\).

Show that Pocklington's theorem leads to the following result of \(\mathrm{E}\). Proth (1878). Let \(n=k \cdot 2^{m}+1\), where \(k\) is odd and \(1 \leq k<2^{m} ;\) if \(a^{(n-1) / 2} \equiv-1(\bmod n)\) for some integer \(a\), then \(n\) is prime.

For what digits \(X\) is \(242628 \times 91715131\) divisible by 3 ?

Find the last digit of \(1999^{1999}\) and the last two digits of \(3^{4321}\).

Find all integers \(n\) that satisfy the equation $$ (n-1)^{3}+n^{3}+(n+1)^{3}=(n+2)^{3} $$ [Hint: Work with the equation obtained by replacing \(n\) by \(k+4 .]\)

Prove that if the odd prime \(p\) divides \(a^{2}+b^{2}\), where \(\operatorname{gcd}(a, b)=1\), then \(p=1(\bmod 4)\).

Prove that the sum $$ 299+2999+29999+\cdots+29999999999999 $$ is divisible by 12 .

Use Pell's equation to show that there are infinitely many integers that are simultaneously triangular numbers and perfect squares.