Solving Congruences and Modular Arithmetic

To convert all numbers to fractions

To ensure all numbers are negative

To isolate X on one side

To find the greatest common divisor

Multiply all terms by the modulus

Find the modulus

Convert to a fraction

Isolate X

Multiplication

Addition

Subtraction

Division

X ≡ 3 (mod 5)

X ≡ 1 (mod 5)

X ≡ 2 (mod 5)

X ≡ 0 (mod 5)

Because 4 is less than 5

Because 8 is a multiple of 5

Because the greatest common divisor is 1

Because 4 is a multiple of 5

It determines the modulus

It allows division in the equation

It converts numbers to fractions

It ensures all numbers are positive

Because 2 is not divisible by 4

Because 4 is greater than 5

Because 2 is a fraction

Because 4 is not a whole number

-1

0

1

4

Multiply by zero

Ignore them

Divide by the modulus

Convert them to positive by adding the modulus

X ≡ 1 (mod 5)

X ≡ 2 (mod 5)

X ≡ -2 (mod 5)

X ≡ -1 (mod 5)