Which integers are congruent to $51 \pmod{14}$?

Question

Which integers are congruent to $51$ mod $14$?

a. 15
b. -19
c. -9
d. 9
e. 121
f. 135

Mod is new to me... but I know $51 = 14 \times 3 + 9$. This in return $= 9 \pmod{14}$.

So the numbers would be: $9, 23, 37, 51, 65, 79, 93, 107, 121, 135, 143, \ldots$

I chose d. ($9$), e. ($121$), and f. ($135$) as the answers on my test... but it appears I missed a point. What did I miss here? I either forgot a number or added an extra number that is not correct.

Thank you!

You have to subtract multiples of $14$ from $9$ as well as adding — Mark Bennet, Oct 31 '17 at 19:52
@MarkBennet Omg duh!! Thank you!! First time doing this type of problem, sorry. So -19 is what I'm missing :D — I'm Hungry, Oct 31 '17 at 19:54

score 1 · Answer 1 · answered Oct 31 '17 at 19:52

1

Since $-19 = 14 \cdot (-2) + 9$, it is congruent to $51$ modulo $14$.

answered Oct 31 '17 at 19:52

score 1 · Answer 2 · answered Oct 31 '17 at 20:15

For this problem it is best to think.

$x \equiv 9 \mod 14 \iff x = 9 \pm km \iff x-9 = \pm km \iff x -9 $ is a multiple of $14$.

Subtracting $9$ from $a-f$ and dividing by $14$ you get

a. 15: 15- 9 = 6: remainder
b. -19:-19-9 = -28: no remainder (good)
c. -9:-9-9 = -18: remainder
d. 9-9 = 0: no remainder (good)
e. 121-9 = 112: =8*14 so no remainder (good)
f. 135 -9= 126: = 112 + 14 so also no remainder  (good)

Hence... $b$. As well as the ones you got.

Which integers are congruent to $51 \pmod{14}$?

2 Answers2