## Princeton Review Error : Congruences Page 225

Forum for the GRE subject test in mathematics.
Legendre
Posts: 217
Joined: Wed Jun 03, 2009 1:05 am

### Princeton Review Error : Congruences Page 225

Princeton Review, rule 3 :

For any positive integer c, the statement

'a is congruent to b (mod n)'
<=>
'a is congruent to b (mod cn)'

Counter-example :

5 congruent 2 (mod 3)

Let c = 2.

5 congruent 2 (mod 9) is FALSE. Since (5-2) = 3 is not divisible by 9.

hopeful_statguy
Posts: 7
Joined: Sun Sep 13, 2009 5:45 pm

### Re: Princeton Review Error : Congruences Page 225

What edition do you have? In the 3rd edition, rule 3 is:
If $a_1 \equiv b_1 \pmod{n}$ and $a_2 \equiv b_2 \pmod{n}$ , then
$a_1 \pm a_2 \equiv b_1 \pm b_2 \pmod{n}$
$a_1 a_2 \equiv b_1 b_2 \pmod{n}$

kosuke
Posts: 5
Joined: Mon Aug 03, 2009 10:27 pm

### Re: Princeton Review Error : Congruences Page 225

Sorry to bring this post back from the dead, but I'm having trouble figuring out what LeDuc meant to say here... (I think Legendre meant to say rule #4, not 3 on page 225).

Any idea on what LeDuc really means?

kosuke
Posts: 5
Joined: Mon Aug 03, 2009 10:27 pm

### Re: Princeton Review Error : Congruences Page 225

Sorry, nevermind.

I figured out what I was doing wrong.

Legendre
Posts: 217
Joined: Wed Jun 03, 2009 1:05 am

### Re: Princeton Review Error : Congruences Page 225

Sorry I meant rule 4, not 3.

I think he meant that a = b (mod n) then, a = b or b + n or b + 2n or ... or b + (c-1)n (mod cn).

This is clear because for b + cn, b + (c+1)n,... we can mod out the cn and obtain same expression in the list.