# [SOLVED] Congruent Modulo

• Mar 27th 2006, 03:57 PM
rexsi
[SOLVED] Congruent Modulo
I have an assignment due tommorow and I am completely lost I was wondering if anyone could give me a hand hehe.

The two silly questions I am stuck on are:

Find all numbers between 10 and 20 which are congruent to 173 modulo 3

and.....

Find all numbers between 10 and 20 which are congruent to -173 modulo 4

:p
• Mar 27th 2006, 06:25 PM
ThePerfectHacker
Quote:

Originally Posted by rexsi
I have an assignment due tommorow and I am completely lost I was wondering if anyone could give me a hand hehe.

The two silly questions I am stuck on are:

Find all numbers between 10 and 20 which are congruent to 173 modulo 3

and.....

Find all numbers between 10 and 20 which are congruent to -173 modulo 4

:p

You need,
$\displaystyle x\equiv 173 \mod 3$
But,
$\displaystyle 173\equiv 2\mod 3$
Thus, (transitive property of congruences)
$\displaystyle x\equiv 2\mod 3$
But, $\displaystyle 10\leq x\leq 20$.
Thus, $\displaystyle x=11,14,17,20$
-----------
You need,
$\displaystyle x\equiv -173 \mod 4$
But,
$\displaystyle -173\equiv 3\mod 4$
Thus, (transitive property of congruences)
$\displaystyle x\equiv 3\mod 4$
But, $\displaystyle 10\leq x\leq 20$
Thus, $\displaystyle x=11,15,19$

$\displaystyle \mathbb{Q}.\mathbb{E}.\mathbb{D}$