Answer:
2
Explanation:
To find the remainder when dividing a number by 11, we can use the property that states if a number is congruent to another number modulo 11, then their powers will also be congruent modulo 11.
Let's calculate the remainders individually and then subtract them:
Remainder of 3192^2109 when divided by 11:
3192 ≡ 3 (mod 11)
3^2109 ≡ 3^(3 × 703) ≡ (3^3)^703 ≡ 27^703 ≡ 5^703 ≡ 5^(4 × 175 + 3) ≡ (5^4)^175 × 5^3 ≡ 625^175 × 125 ≡ 4^175 × 4 ≡ 4^(4 × 43 + 3) × 4 ≡ (4^4)^43 × 4^3 ≡ 256^43 × 64 ≡ 3^43 × 9 ≡ 3 × 9 ≡ 27 ≡ 5 (mod 11)
Remainder of 3159^2109 when divided by 11:
3159 ≡ 9 (mod 11)
9^2109 ≡ 9^(3 × 703) ≡ (9^3)^703 ≡ 729^703 ≡ 8^703 ≡ 8^(4 × 175 + 3) ≡ (8^4)^175 × 8^3 ≡ 4096^175 × 512 ≡ 1^175 × 3 ≡ 3 (mod 11)
Subtracting the remainders:
5 - 3 ≡ 2 (mod 11)
Therefore, the remainder when dividing 3192^2109 - 3159^2109 by 11 is 2.