Match List-I with List-II

Choose the correct answer from the options given below:

(A)-(II), (B)-(IV), (C)-(I), (D)-(III)

The correct answer is Option (1) → (A)-(II), (B)-(IV), (C)-(I), (D)-(III)

(A) Remainder when $17^{35234}$ is divided by 8:

• First, find the remainder of the base: $17 \div 8 = 2$ with a remainder of 1.
• Using the property of remainders: $17^{n} \equiv 1^{n} \pmod{8}$.
• Since $1^{35234} = 1$, the remainder is 1.
• Match: (A) - (II)

(B) Remainder when 4444 is divided by 9:

• To find the remainder when a number is divided by 9, we sum its digits: $4 + 4 + 4 + 4 = 16$.
• Now, divide the sum by 9: $16 \div 9 = 1$ with a remainder of 7.
• Match: (B) - (IV)

(C) Unit's digit of $(34)^{15} + (34)^{16}$:

• We only need to look at the last digit, which is 4.
• The cyclicity of 4 is: $4^1 = 4$, $4^2 = 16$ (ends in 6), $4^3 = 64$ (ends in 4).
• $4^{\text{odd}}$ ends in 4.
• $4^{\text{even}}$ ends in 6.
• $(34)^{15}$ has an odd power, so its unit digit is 4.
• $(34)^{16}$ has an even power, so its unit digit is 6.
• Sum of unit digits: $4 + 6 = 10$. The unit digit of the result is 0.
• Match: (C) - (I)

(D) Unit digit of $7^4 - 9^3$:

• $7^4$: $7 \times 7 \times 7 \times 7 = 2401$. The unit digit is 1.
• $9^3$: $9 \times 9 \times 9 = 729$. The unit digit is 9.
• Subtraction: $1 - 9 ⇒11 - 9$ (borrowing from the next place value) = 2.
• Match: (D) - (III)