1. ## [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

2. 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

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

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