To solve for the remainder of [tex]\( n = (\underbrace{555 \ldots 555}_{30185 \text{ times}})^2 - (\underbrace{999 \ldots 999}_{50069 \text{ times}})^2 \)[/tex] when divided by 11, we utilize properties of numbers and modular arithmetic.
First, consider the number [tex]\( a = \underbrace{555 \ldots 555}_{30185 \text{ times}} \)[/tex]. This is a number composed entirely of 30185 digits of 5. Similarly, let [tex]\( b = \underbrace{999 \ldots 999}_{50069 \text{ times}} \)[/tex], composed entirely of 50069 digits of 9.
### Step 1: Examine [tex]\( a \)[/tex] modulo 11
To find the remainder of [tex]\( a \)[/tex] modulo 11, we use the property of congruence with alternating sum of digits. That is, [tex]\( a \equiv (\text{sum of digits} \mod 11) \)[/tex].
For [tex]\( a = 555 \ldots 555 \)[/tex]:
- The digits are all 5.
- Total number of digits = 30185.
Sum of the digits of [tex]\( a \)[/tex]:
[tex]\[ 30185 \times 5 = 150925 \][/tex]
Calculating [tex]\( 150925 \mod 11 \)[/tex]:
[tex]\[ 150925 \equiv 150925 \pmod{11} \][/tex]
To simplify:
[tex]\[ 1 + 5 + 0 + 9 + 2 + 5 = 22 \][/tex]
[tex]\[ 22 \div 11 = 2 \][/tex]
[tex]\[ 22 \equiv 0 \pmod{11} \][/tex]
Therefore:
[tex]\[ 150925 \equiv 0 \pmod{11} \][/tex]
Thus:
[tex]\[ a \equiv 0 \pmod{11} \][/tex]
### Step 2: Examine [tex]\( b \)[/tex] modulo 11
Similarly, for [tex]\( b = 999 \ldots 999 \)[/tex]:
- The digits are all 9.
- Total number of digits = 50069.
Sum of the digits of [tex]\( b \)[/tex]:
[tex]\[ 50069 \times 9 = 450621 \][/tex]
Calculating [tex]\( 450621 \mod 11 \)[/tex]:
[tex]\[ 450621 \equiv 450621 \pmod{11} \][/tex]
To simplify:
[tex]\[ 4 + 5 + 0 + 6 + 2 + 1 = 18 \][/tex]
[tex]\[ 18 \div 11 = 1 \][/tex]
[tex]\[ 18 \equiv 7 \pmod{11} \][/tex]
Therefore:
[tex]\[ 450621 \equiv 7 \pmod{11} \][/tex]
Thus:
[tex]\[ b \equiv 7 \pmod{11} \][/tex]
### Step 3: Examine [tex]\( n = a^2 - b^2 \mod 11 \)[/tex]
We now calculate [tex]\( a^2 - b^2 \mod 11 \)[/tex].
[tex]\[ a \equiv 0 \pmod{11} \][/tex]
[tex]\[ b \equiv 7 \pmod{11} \][/tex]
So:
[tex]\[ a^2 \equiv 0^2 \equiv 0 \pmod{11} \][/tex]
[tex]\[ b^2 \equiv 7^2 \equiv 49 \pmod{11} \][/tex]
[tex]\[ 49 \div 11 = 4 \][/tex]
[tex]\[ 49 \equiv 5 \pmod{11} \][/tex]
Thus:
[tex]\[ n = a^2 - b^2 \][/tex]
[tex]\[ n \equiv 0 - 5 \equiv -5 \pmod{11} \][/tex]
Since -5 is equivalent to 6 modulo 11 (because [tex]\(-5 + 11 = 6\)[/tex]):
[tex]\[ n \equiv 6 \pmod{11} \][/tex]
So, the remainder when [tex]\( n \)[/tex] is divided by 11 is [tex]\( \boxed{6} \)[/tex].
