Match List-I with List-II

List-I	List-II
(A) Remainder when $17^{35234}$ is divided by 8	(I) 0
(B) Remainder when 4444 is divided by 9	(II) 1
(C) Unit's digit of $(34)^{15} + (34)^{16}$	(III) 2
(D) Unit digit of $7^4 - 9^3$	(IV) 7

Choose the correct answer from the options given below:

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

When finding remainders, simplify the base first.

$17 = 8 \times 2 + 1$

So 17 leaves remainder 1 when divided by 8.

The problem becomes finding the remainder when $1^{35234}$ is divided by 8.

$1^{35234} = 1$

Remainder $= 1$ → (II)

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(B) Remainder when 4444 is divided by 9

A number and its digit sum have the same remainder when divided by 9.

Sum of digits of 4444:

$4 + 4 + 4 + 4 = 16$

$1 + 6 = 7$

Remainder $= 7$ → (IV)

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

Only the unit digit of 34 matters, which is 4.

Pattern of powers of 4:

$4^1 = 4$ (unit digit: 4)

$4^2 = 16$ (unit digit: 6)

$4^3 = 64$ (unit digit: 4)

$4^4 = 256$ (unit digit: 6)

The pattern alternates: 4, 6, 4, 6...

$(34)^{15}$ has odd power → unit digit $= 4$

$(34)^{16}$ has even power → unit digit $= 6$

$4 + 6 = 10$

Unit digit $= 0$ → (I)

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

For $7^4$:

$7^1 = 7$

$7^2 = 49$ (unit digit: 9)

$7^3 = 343$ (unit digit: 3)

$7^4 = 2401$ (unit digit: 1)

For $9^3$:

$9^3 = 729$ (unit digit: 9)

The subtraction: $1 - 9$

Since $1 < 9$, this becomes $(10 + 1) - 9$

$= 11 - 9$

$= 2$

Unit digit $= 2$ → (III)

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Final Matching:

(A) → (II)

(B) → (IV)

(C) → (I)

(D) → (III)