# modular arithmetic question

#### Coefficient

##### New member
Hi,
I'm stuck on this question: solve for x (x is an integer) 7x = -19 mod(9) .

#### Dr.Peterson

##### Elite Member
I would start by rewriting -19 as a smaller number (mod 9). What is -19 congruent to? That is not necessary, but will make things easier.

Then you can just try numbers 1 through 8 to see if any yield a product congruent to the number you found.

• topsquark

#### Coefficient

##### New member
i tried this :

7x ≡ -19
7x ≡ -10
7x ≡ -1
7x ≡ 8

But i couldn't understand how this would help me because the next question is to deduce the solutions of the equation : 7x-9y= -19

Thanks.

#### topsquark

##### Full Member
i tried this :

7x ≡ -19
7x ≡ -10
7x ≡ -1
7x ≡ 8

But i couldn't understand how this would help me because the next question is to deduce the solutions of the equation : 7x-9y= -19

Thanks.
Note that 9y = 0 (mod 9). So the solutions for 7x - 9y = -19 (mod 9) will be the solutions for 7x = -19 (mod 9) and y = anything.

-Dan

• Jomo and Coefficient

#### Coefficient

##### New member
Note that 9y = 0 (mod 9). So the solutions for 7x - 9y = -19 (mod 9) will be the solutions for 7x = -19 (mod 9) and y = anything.

-Dan
Thank you

#### Jomo

##### Elite Member
What does 7x ≡ 8 mean? If I remember correctly it means that 7x-8 = 9k, where k is an integer. So solve for x.

7x = 9k+8, then x= (9k+8)/7. Now find k so that (9k+8)/7 is an integer.

• Coefficient

#### Jomo

##### Elite Member
So what answer did you get??