Confused about modular notations

I am little confused about the notations used in two articles at wikipedia.

According to the page on Fermat Primality test

$a^{p-1}\equiv 1 \pmod{m}$ means that when $a^{p-1}$ is divided by $m$, the remainder is 1.

And according to the page on Modular Exponential

$c \equiv b^e \pmod{m}$ means that when $b^e$ is divided by $m$, the remainder is $c$.

Am I interpreting them wrong? or both of them are right? Can someone please clarify them to me?

Update: So given an equation say $x \equiv y \pmod{m}$, how do we interpret it? $y$ divided by $m$ gives $x$ as remainder or $x$ divided by $m$ gives $y$ as remainder?

Answer

x ≡ y (mod m)

is equivalent to:

x divided by m gives the same remainder as y divided by m

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So, in your first example:

a^(p-1) ≡ 1 (mod m)

is equivalent to:

a^(p-1) divided by m gives the same remainder as 1 divided by m

But 1 divided by m gives remainder 1, so the above becomes:

a^(p-1) divided by m gives remainder 1

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In your second example:

c ≡ b^e (mod m)

is equivalent to:

c divided by m gives the same remainder as b^e divided by m

But (assuming that 0 <=c < m) we have c divided by m gives remainder c, so the above becomes:

c is the remainder that b^e divided by m gives.

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As it seems that the notation may be confusing, let me note that

x ≡ y (mod m)

should be read as:

[ x ≡ y ] (mod m)         --- the (mod m) is applied to the whole congruence

and not as:

x ≡ [ y (mod m) ]         --- and not just to the right part.