Congruence Examples for High School

The value of the operation [12]+[25] in ${\displaystyle Z_7}$ and to write the answer in the form [r] with ${\displaystyle 0\le r<m}$.

To determine: The smallest nonnegative integer x that satisfies the given system of congruences.
${\displaystyle x\equiv 1003\pm od\{17,369\}}$
${\displaystyle x\equiv 2974\pm od\left\{5472\right\}}$

Find a complete set of mutually incogruent solutions.
${\displaystyle 7x\equiv 5\pm od\left\{11\right\}}$

The value of the operation $[9]+[8]\in$ ${\displaystyle Z_{13}}$ and to write the answer in the form [r] with ${\displaystyle 0\le r<m}$.

The value of the operation [5][12] in ${\displaystyle Z_8}$ and to write the answer in the form [r] with ${\displaystyle 0\le r<m}$.

Find a complete set of mutually incogruent solutions.
${\displaystyle 9x\equiv 12\pm od\left\{15\right\}}$

To determine: The smallest nonnegative integer x that satisfies the given system of congruences.
${\displaystyle x\equiv 1\pm od4}$
${\displaystyle x\equiv 8\pm od9}$

What other information do you need in order to prove the triangles congruent using the SAS Congruence Postulate?

A)${\displaystyle \mathrm{\angle }BAC\stackrel{\sim }{=}\mathrm{\angle }DAC}$
B)${\displaystyle \overline{AC}\stackrel{\sim }{=}\overline{BD}}$
C)${\displaystyle \mathrm{\angle }BCA\stackrel{\sim }{=}\mathrm{\angle }DCA}$
D)${\displaystyle \overline{AC}\stackrel{\sim }{=}\overline{BD}}$

Show that the congruence ${\displaystyle 8x^2-x+4\equiv 0\left(\text{mod}9\right)}$ has no solution by reducing to the form ${\displaystyle y^2\equiv a\left(\text{mod}9\right)}$

For the following statement, either prove that they are true or provide a counterexample:
Let a, b, c, d, ${\displaystyle m\in Z}$ such that c, ${\displaystyle d\ge 1}$ and m > 1. If ${\displaystyle a\equiv b\left(\text{mod}m\right)}$ and
${\displaystyle c\equiv d\left(\text{mod}m\right)}$, then ${\displaystyle a^c\equiv b^d\left(\text{mod}m\right)}$

The value of the operation [9][6] in ${\displaystyle Z_{10}}$ and to write the answer in the form [r] with ${\displaystyle 0\le r<m}$.

Solve the set of congruences
${\displaystyle 2x\equiv 1\left(\text{mod}5\right)}$
${\displaystyle x\equiv 3\left(\text{mod}4\right)}$

For each of the following congruences, find all integers N, with N>1, that make the congruence true.
$10\equiv 5(modN)$.

Solve the following systems of congruences.
$x\equiv 4(mod7)$

Find the solution of congruence:
${\displaystyle x^{43}\equiv 47\left(\text{mod}713\right)}$

Justify Conclusions take a conjecture about the sum of th interior angles of a quadrilateral. Justify your reasoning.

A Counterexample to the statement, If ${\displaystyle a\equiv b\pm odm}$ and ${\displaystyle c\equiv d\pm odm}$, where a,b,c,d, and m are integers with c and d positive and ${\displaystyle m\ge 2}$, then ${\displaystyle a^c\equiv b^d\pm odm}$.

Find the triangle congruence (SSS SAS AAS HL)

Given: bar(SQ) and bar(PR) bisect each other.

Solve the following linear congruence: $17xcongruence3(mod210)$

Find the least positive solution for the following congruence: $41x=125(mod660)$.