Congruence Examples for High School

To determine:The measure of angle 6.
Given:
ABDC is a rectangle.

${\displaystyle m\mathrm{\angle }1=38}$

How are the measures of the angles related when parallel lines are cut by a transversal?

The value of the operation [8][11] in ${\displaystyle Z_5}$ and to write the answer in the form [r] with ${\displaystyle 0\le r<m}$.

Whether ${\displaystyle p\equiv q\pm odm}$, when p=29,q=-34, and m=7.

The value of the operation [3][20] in ${\displaystyle Z_{11}}$ and to write the answer in the form [r] with ${\displaystyle 0\le r<m}$.

The value of the expression [5]+[10] in ${\displaystyle Z_{11}}$ and to write the answer in the form [r] with ${\displaystyle 0\le r<m}$.

Whether ${\displaystyle p\equiv q\pm odm}$ when p = 91, q=37, and m =9.

To find:
The smallest positive integer such that if we divide it by three, the remainder is 2. if we divide it by five, the remainder is 3. if we divide it by seven, the remainder is 2.

To determine: The smallest nonnegative integer x that satisfies the given system of congruences.
${\displaystyle x\equiv 1\pm od\left\{4\right\}}$
${\displaystyle x\equiv 8\pm od\left\{9\right\}}$
${\displaystyle x\equiv 10\pm od\left\{25\right\}}$

To determine: The smallest nonnegative integer x that satisfies the given system of congruences.
${\displaystyle x\equiv 3\pm od\left\{1917\right\}}$
${\displaystyle x\equiv 75\pm od\left\{385\right\}}$

A Counterexample to the statement, If $ac\equiv bc(\mathrm{mod}m)$, where a,b,c, and m are integers with ${\displaystyle m\ge 2}$, then $a\equiv b(\mathrm{mod}m)$.

The value of the operation [8][7] in ${\displaystyle Z_6}$ and to write the answer in the form [r] with ${\displaystyle 0\le rm}$.

To determine:The check digit of an ISSN that has these initial 7 digits 1553-734

The value of the expression [9]+[11] in ${\displaystyle Z_{15}}$ and to write the answer in the form [r] with ${\displaystyle 0\le r<m}$.

The value of the operation ${\displaystyle {\left[11\right]}^8}$ in ${\displaystyle Z_5}$ and to write the answer in the form [r] with ${\displaystyle 0\le m}$.

To determine: The smallest nonnegative integer x that satisfies the given system of congruences.
${\displaystyle x\equiv 6\pm od8}$
${\displaystyle x\equiv 17\pm od\left\{25\right\}}$

Solve the following congruence.
${\displaystyle x^2\equiv 7b\text{mod}29}$

The value of the operation [16]+[9] in ${\displaystyle Z_6}$ 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.
${\displaystyle x\equiv 3\pm od5}$
${\displaystyle x\equiv 7\pm od8}$

To find:
The smallest the smallest positive integer such that if we divide it by two, three and four, the remainder is one but seven divides the number evenly.