Discrete Math: Number Theory and Modular Arithmetic

Solve the linear congruence: $2x \equiv 13 \mod 7$ after dividing both sides by 2.

Solve the linear congruence: $5x \equiv 3 \mod 29$ and provide the solution in a positive form less than 29.

Solve the linear congruence: $3x \equiv 14 \mod 2$ and find all solutions in the least residue system.

Solve the linear congruence: $2x \equiv 5 \mod 7$ and find all three solutions using the parametric form.

Solve the linear congruence equation $4x \equiv 8 \pmod{5}$

$4x \equiv 2 \, \pmod{5}$

$4x \equiv 3 \pmod{5}$

If $a$ divides $b$ and $a$ divides $c$, then $a$ divides $b + c$.

If $a$ divides $b$ and $b$ divides $c$, then $a$ divides $c$.

If $a$ divides $c$ and $b$ divides $d$, then $ab$ divides $cd$; verify if this is true.

If $a$ divides $b$ and $b$ divides $c$, then $a$ divides $c$.

If $a$ divides $b$, then $c \times a$ divides $c \times b$.

If $a$ divides $b$ and $a$ divides $c$, then for all integers $x$ and $y$, $a$ divides $bx + cy$.

If $a$ divides 1, then $a = \pm 1$.

Let $R$ be a relation on the set of integers defined by $a \, R \, b$ if and only if $a - b$ is an integer. Prove that $R$ is an equivalence relation.

Find the GCD of 12, 33 using the method shown in the video.

Find the GCD of 25, 150 using the method shown in the video.

Find the GCD of 13, 31 using the method shown in the video.

Find the GCD of 750 and 900 using the Euclidean algorithm.

Find the GCD of 252, 105 using the Euclidean algorithm.