Using 10^n = x (mod 9).
Given that
d=2935107
=2*10^6 + 9*10^5 + 3*10^4 + 5*10^3 + 1*10^2 + 0*10^1 + 7^10^0
Then d = kx (mod 9).
What is the smallest positive value of k?
well, there may be something missing here, but taking it as written,
since 10^n = 1 (mod 9)
d = 2+0+3+5+1+0+7 (mod 9) = 0 mod 9
k=9 would fit the bill, since k=0 is not positive.
To find the smallest positive value of k such that d = kx (mod 9), we need to calculate the value of x (mod 9) first.
The given equation is 10^n = x (mod 9).
We can rewrite this equation as 10^n - x = 0 (mod 9).
Now let's calculate the value of x (mod 9):
For any integer a, a (mod 9) is the remainder obtained when dividing a by 9.
Therefore, we calculate x (mod 9) as follows:
x ≡ 10^n (mod 9)
x ≡ (9+1)^n (mod 9)
x ≡ 1^n (mod 9)
x ≡ 1 (mod 9)
So, we have x ≡ 1 (mod 9).
Now we substitute x = 1 into the equation d = kx (mod 9):
d ≡ k(1) (mod 9)
d ≡ k (mod 9)
To find the smallest positive value of k, we need to consider the remainder when d is divided by 9.
Given that d = 2935107, we calculate the remainder of d divided by 9:
d ≡ 2935107 (mod 9)
d ≡ (2*10^6 + 9*10^5 + 3*10^4 + 5*10^3 + 1*10^2 + 0*10^1 + 7^10^0) (mod 9)
d ≡ (2 + 9 + 3 + 5 + 1 + 0 + 7) (mod 9)
d ≡ 27 (mod 9)
d ≡ 0 (mod 9)
Therefore, we have the equation:
0 ≡ k (mod 9)
The smallest positive value of k is k = 9, which satisfies the equation.
So, the smallest positive value of k is 9.