8 ≡ 6 (mod 2)? Is the (mod 2) applied to both sides?

Question

8 ≡ 6 (mod 2)

Is this True or False? And why?

Is the (mod 2) applied to both sides of the logical equivalence?

Answer 1

In mathematics, "(mod 2)" is usually not a function. It does not "apply" to a side of an equation. It is a description of the entire equation: that the equation should be taken "(mod 2)".

Depending on the vocabulary you are comfortable with, this means one of the following (equivalent) things:

• The equation holds "up to multiples of 2".
• There is a $k$ in the integers, so that adding $2k$ to one side of the equation makes it true.
• If you subtract one side of the equation from the other, the result is divisible by 2.
• The equation should be read as being about elements of $\mathbb Z/2\mathbb Z$.

For example, the following are all true: $$ 8 = 0 \pmod2\\ 15 = -9 \pmod 2\\ 0 = 18 \pmod2\\ 20 = 20 \pmod2 $$ To emphasize that these are (usually) not exactly equalities of integers, sometimes additionally the equals symbol $=$ is replaced by another, usually $\equiv$.