0

I know that a duplicate of this question is posted here ,but i am still posting this because the post contain only hint.I solved this using myself ,and i want to verify it.

Question

Suppose that $a$ and $b$ are integers, $a ≡ 4 (mod 13)$, and $b ≡ 9 (mod 13)$.
Find the integer $c$ with $0 ≤ c ≤ 12$ such that

  1. $c ≡ 9\,\,a (\text {mod} \,\,13)$

My attempt

Given

$\Rightarrow$ $a ≡ 4 (mod 13)$

we can write it using symmetric as,

$\Rightarrow$ $4 ≡ a (mod 13)$

$\Rightarrow$ $b ≡ 9 (mod 13)$.

We can write above as

$\Rightarrow$ $4*b ≡ 9a (mod 13)$.

and our question is

$c ≡ 9\,\,a (\text {mod} \,\,13)$

I am stuck here , any way to move forward?

Community
  • 1
laura
  • 2,530

2 Answers2

2

Just compute, using $a\equiv 4\bmod 13$, that $$ 9a\equiv 9\cdot 4=36\equiv 10\bmod 13. $$ hence we have $c=10$. We do not need to know $b$, by the way.

Dietrich Burde
  • 130,978
  • we can choose $a$ as $4+13$ ..why we choosen $a=4$? so that $c$ lies within the asked range? – laura May 31 '17 at 12:29
  • If we write $\equiv$, then $4\equiv 4+13k$ for all $k$, so this makes no difference. And I wrote $9a\equiv 9\cdot 4$ and not $9a=9\cdot 4$. – Dietrich Burde May 31 '17 at 12:32
  • Thank you. Is it legal to replace ≡ with = sign? How you replace c≡9a(mod13) with c=9a(mod13) – Avv Mar 16 '21 at 03:35
  • Almost. We have that $c \equiv 9a \bmod 13$ is equivalent to the equation $c=9a$ in the ring $\Bbb Z/13$. – Dietrich Burde Mar 16 '21 at 08:55
2

hint

$$9a\equiv 36 \equiv -3 \; (13) $$

$$\implies c\equiv -3 \;(13 ) $$ $$\implies c=10$$

hamam_Abdallah
  • 62,951