7¹ = 7,
7² = 49,
7³ = 343,
7⁴ = 2401
3¹ = 3,
3² = 9,
3³ = 27,
3⁴ = 81
i.e. the digit at unit’s place gets repeated after power 4. Unit 6 remains same for any power.

Correct Option: D

7¹ = 7,
7² = 49,
7³ = 343,
7⁴ = 2401
3¹ = 3,
3² = 9,
3³ = 27,
3⁴ = 81
i.e. the digit at unit’s place gets repeated after power 4. Unit 6 remains same for any power.
∴  Required unit’s digit
= Unit’s digit in the product of 7³ × 6 × ¹ = 4

2¹ = 2
2² = 4
2³ = 8
2⁴ = 16
2⁵ = 32
Now As we know that,
Unit place of 2¹ = 2
Unit place of 2² = 4
Unit place of 2³ = 8
Unit place of 2⁴ = 6
Unit place of 2⁵ = 2
i.e. 2 repeats itself after the index 4.

Correct Option: A

2¹ = 2
2² = 4
2³ = 8
2⁴ = 16
2⁵ = 32
Now As we know that,
Unit place of 2¹ = 2
Unit place of 2² = 4
Unit place of 2³ = 8
Unit place of 2⁴ = 6
Unit place of 2⁵ = 2
i.e. 2 repeats itself after the index 4.
∴  Unit digit in the product of (122)¹⁷³
= Unit digit in (122)¹ = 2
(1 = remainder when 173 is divided by 4).

Unit’s digit in the expansion of (22)²³ = Unit’s digit in the expansion of (2)²³
2¹ = 2
2² = 4
2³ = 8
2⁴ = 16
2⁵ = 32

Correct Option: C

Unit’s digit in the expansion of (22)²³ = Unit’s digit in the expansion of (2)²³
2¹ = 2
2² = 4
2³ = 8
2⁴ = 16
2⁵ = 32
Now As we know that,
Unit place of 2¹ = 2
Unit place of 2² = 4
Unit place of 2³ = 8
Unit place of 2⁴ = 6
Unit place of 2⁵ = 2
i.e. 2 repeats itself after the index 4.
On dividing 23 by 4, remainder = 3
∴  Unit’s digit in (2)²³ = Unit’s digit in (2)³ = 8

3¹ = 3;
3² = 9;
3³ = 27;
3⁴ = 81;
3⁵ = 343 ; ...........................................
We can write 3⁴⁰ as below,
3⁴⁰ = 3⁴ x 3⁴ x ........ 10 times
Unit place digit for 3⁴⁰ = Unit place Digit of 3⁴ x Unit place Digit of 3⁴ x.......... 10 times

Correct Option: A

3¹ = 3;
3² = 9;
3³ = 27;
3⁴ = 81;
3⁵ = 343 ; ...........................................
We can write 3⁴⁰ as below,
3⁴⁰ = 3⁴ x 3⁴ x ........ 10 times
Unit place digit for 3⁴⁰ = Unit place Digit of 3⁴ x Unit place Digit of 3⁴ x.......... 10 times
Unit place digit for 3⁴⁰ = 1 x 1 x............. 10 times
∴  Unit’s digit in 3⁴⁰ = 1

Unit digit in (264)⁴ 4 × 4 × 4 × 4 is 6
∴  Unit digit in (264)¹⁰⁰ is also 6.
Now, (264)¹⁰² = (264)¹⁰⁰ × (264)²
= (Unit digit 6) × (Unit digit 6) = 36
∴  Unit digit is 6

Correct Option: A

Unit digit in (264)⁴ 4 × 4 × 4 × 4 is 6
∴  Unit digit in (264)¹⁰⁰ is also 6.
Now, (264)¹⁰² = (264)¹⁰⁰ × (264)²
= (Unit digit 6) × (Unit digit 6) = 36
∴  Unit digit is 6
Similarly

Therefore, the unit digit in (264)¹⁰² + (264)¹⁰³ is 6 + 4 = 10 i.e. 0.