# [SOLVED] congruent modulo n

• Dec 8th 2009, 06:21 AM
sudeepmansh
[SOLVED] congruent modulo n
For the statement "a is congruent to b modulo n"
Which of the following definition is correct.

a) a - b is divisible by n.

b) a - b is a multiple of n.

or whether both are acceptable.

Whether "a is congruent to a mod 0" is true?

a - a = 0 is divisible by 0 is wrong. But 0 is a multiple of 0 is correct. So whether to consider
the definition (a) or definition (b).
• Dec 8th 2009, 06:47 AM
dedust
for every $a \in \mathbb{R}$, $a = a + k(0)$, $k \in \mathbb{N}$
so $a \equiv a ~mod ~0$.

for this case, just use definition (b)

regards
• Dec 8th 2009, 10:53 AM
emakarov
Quote:

Originally Posted by sudeepmansh
For the statement "a is congruent to b modulo n"
Which of the following definition is correct.

a) a - b is divisible by n.

b) a - b is a multiple of n

The correct answer: in mathematics, it is not customary to argue about definitions.

That said, both MathWorld and Planet Math use (a), and Planet Math specifies that $n\ne 0$. If I am not mistaken, both definitions are equivalent if $n\ne 0$.

I agree with the OP's assessment of "a is congruent to a mod 0".