GATE Electrical 2023 | GA Question: 9

Question

The digit in the unit's place of the product $3^{999} \times 7^{1000}$ is _________.

• A. $7$
• B. $1$
• C. $3$
• D. $9$

Answer 1

The pattern of the unit's digits of powers of $3:$

1. $3^1 = 3$
2. $3^2 = 9$
3. $3^3 = 7$
4. $3^4 = 1$

The pattern repeats every $4$ powers of $3.$

The pattern of the unit's digits of powers of $7:$

1. $7^1 = 7$
2. $7^2 = 9$
3. $7^3 = 3$
4. $7^4 = 1$

The pattern repeats every $4$ powers of $7.$

Now, Unit digit of $3^{999} \times 7^{1000} = \underbrace{3^{4{(249)}}}_{\text{Unit digit = 1}} \times 3^3 \times \underbrace{7^{4(250)}}_{\text{Unit digit = 1}} = 1 \times 7 \times 1 = 7$

So, the unit's digit of $3^{999} \times 7^{1000}$ is $7.$

Correct Answer: A