# modular arithmetic

#### Coefficient

##### New member
Hi,

I am working on this exercise that says:
1- Decompose 1995 into prime factors.
2- Find all pairs (x,y) ∈ ℕ² so that : x+7y = 1995 and gcd(x,y) = 19

Here is what i've come to so far :

1995 = 3*5*7*19

x=19*x′ and y=19y′ so that gcd(x′,y′)=1
by pluggin this in my equation :

19x′+7*19y′=1995
19(x′+7y′)=1995
x′+7y′=105
x′=105-7y′
x′=7(15-y′)
so i concluded that: x′=7 and 15-y′=1
so y′=14 then x=7*19=133 and y=14*19=266
But after looking at the solution i found out i was wrong, because there is 6 pairs of numbers that satisfy the previous conditions.
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