Congruence $320 \equiv 1 (\text{mod }x)$

Question

I have the following congruence $320 \equiv 1 (\text{mod }x)$

And the question is : find all the modulos $x$ that make this congruence true.

@GregoryGrant not only has, but it is also sufficient for $x$ to divide $319$. i.e. $x$ satisfies the conditions iff $x$ divides $319$. — user26486, May 17 '15 at 14:03

score 3 · Accepted Answer · edited May 17 '15 at 09:18

3

Well basically the definition of $$a \equiv b (\text{mod }n)$$ is that n should divide $(a-b)$ So according to your question $x$ should divide $320-1$ that is $319$. It would be $11$ and $29$

Since you want all the $x$'s then it would be $11$, $29$ and $319$

edited May 17 '15 at 09:18

wythagoras

25,026

answered May 17 '15 at 07:28

and 319 too.... – Honza Brabec May 17 '15 at 07:40
Yes I saw that thanks@HonzaBrabec – May 17 '15 at 07:41
And $1$. And, if you allow negative moduli (some people do), four more. – André Nicolas May 17 '15 at 07:43
@AndréNicolas 320 mod 1 is 0. – Honza Brabec May 17 '15 at 07:45
@HonzaBrabec 0 is congruent to 1 (mod 1). – wythagoras May 17 '15 at 07:46
@wythagoras Thanks :) – Honza Brabec May 17 '15 at 07:47
I dont know.... i just provided few answers @AndréNicolas i dont want to tell all answers – May 17 '15 at 07:47
Yes i know 1 will be there@AndréNicolas – May 17 '15 at 07:49

score 0 · Answer 2 · answered May 17 '15 at 07:33

0

$320 \equiv 1 (\text{mod }x) \rightarrow 320-1 \equiv 0 (\text{mod }x) \rightarrow 319 \equiv 0 (\text{mod }x) \rightarrow x=11$ or $29$, since that only works for the divisors.

answered May 17 '15 at 07:33

Atvin

3,392

score 0 · Answer 3 · answered May 17 '15 at 07:49

$320 \equiv 1 \pmod{x} \implies x \mid 320 -1 =319$ and hence there exists an integer $k$ such that $$xk = 319$$

Now should find all pairs $x,k$ that when you multiply them together you get $319$

$1 \times 319$

$29 \times 11$

$-1 \times -319$

$-29 \times -11$

any one of these 8 numbers can work for $x$

Basically you get the prime factorization for $319$ and then you add to your list the negatives of these numbers plus the number itself plus 1, that will give you a complete list for the divisors of $319$ or generally any number.

Congruence $320 \equiv 1 (\text{mod }x)$

3 Answers3