Number Theory
Elementary Number Theory Problems 4.2 Solution (David M. Burton's 7th Edition) - Q6 Paid Members Public
My Solution for "For $n \geq 1$, use congruence theory to establish each of the following divisibility statements: (a) $7 \mid 5^{2n} + 3 \cdot 2^{5n-2}$. (b) $13 \mid 3^{n+2} + 4^{2n+1} $. (c) $27 \mid 2^{5n+1} + 5^{n+2}$. (d) $43 \mid 6^{n+2} + 7^{2n+1}$."
Elementary Number Theory Problems 4.2 Solution (David M. Burton's 7th Edition) - Q5 Paid Members Public
My Solution "Prove that the integer $53^{103} + 103^{53}$ is divisible by $39$, and that $111^{333}$ + $333^{111}$ is divisible by $7$."
Elementary Number Theory Problems 4.2 Solution (David M. Burton's 7th Edition) - Q4 Paid Members Public
My Solution for "(a) Find the remainders when $2^{50}$ and $41^{65}$ are divided by $7$. (b) What is the remainder when the following sum is divided by $4$? $$1^5 + 2^5 + 3^5 + \cdots + 99^5 + 100^5$$"
Elementary Number Theory Problems 4.2 Solution (David M. Burton's 7th Edition) - Q3 Paid Members Public
My Solution for "If $a \equiv b \pmod n$, prove that $gcd(a, n) = gcd(b, n)$."
Elementary Number Theory Problems 4.2 Solution (David M. Burton's 7th Edition) - Q2 Paid Members Public
My Solution for "Give an example to show that $a^{2} \equiv b^{2} \pmod n$ need not imply that $a \equiv b \pmod n$."
Elementary Number Theory Problems 4.2 Solution (David M. Burton's 7th Edition) - Q1 Paid Members Public
My Solution for "Prove each of the following assertions: (a) If $a \equiv b \pmod n$ and $m \mid n$, then $a \equiv b \pmod m$. (b) If $a \equiv b \pmod n$ and $c > 0$, then $ca \equiv cb \pmod {cn}$. (c) If $a \equiv b \pmod n$ and the integers $a$, $b$, $n$ are all divisible by $d > 0$..."
Elementary Number Theory Problems 3.3 Solution (David M. Burton's 7th Edition) Paid Members Public
My solutions for Burton's Elementary Number Theory Problems 3.3 (7th Edition)
Elementary Number Theory Problems 3.3 Solution (David M. Burton's 7th Edition) - Q28 Paid Members Public
My Solution for "(a) If $n > 1$, show that $n!$ is never a perfect square. (b) Find the values of $n \geq 1$ for which $$n! + (n + 1)! + (n + 2)!$$ is a perfect square. [Hint: Note that $n! + (n + 1)! + (n + 2)! = n!(n + 2)^{2}$ .]"
