Systems Of Linear Equations

Step 1: Find the unit digit of $3%5E%7B999%3C%2Fstrong%3E$ .}
The powers of $3$ follow a cycle of unit digits: $3%2C%209%2C%207%2C%201$ . This repeats every 4 terms. Now, $999%20%5Cdiv%204$ leaves a remainder of $3$ . Hence, $3%5E%7B999%7D$ will have the same unit digit as $3%5E3$ , which is $7$ .
Step 2: Find the unit digit of $7%5E%7B1000%3C%2Fstrong%3E$ .}
The powers of $7$ follow a cycle of unit digits: $7%2C%209%2C%203%2C%201$ . This repeats every 4 terms. Now, $1000%20%5Cdiv%204$ leaves a remainder of $0$ . Hence, $7%5E%7B1000%7D$ will have the same unit digit as $7%5E4$ , which is $1$ .
Step 3: Multiply the unit digits.
The unit digits are $7%20%5Ctimes%201%20%3D%207$ .
Step 4: Verify carefully.
Wait — but let us double-check: - $3%5E%7B999%7D$ ends in $7$ . - $7%5E%7B1000%7D$ ends in $1$ . So, product’s unit digit $%3D%207%20%5Ctimes%201%20%3D%207$ . Therefore, the correct unit digit is $7$ , not $1$ . Final Answer: $%5Cboxed%7B7%7D$

High-Yield Trend

Chapter Questions
2 MCQs

Official Solution

Official Solution

Systems Of Linear Equations

High-Yield Trend

Chapter Questions 2 MCQs

Official Solution

Official Solution

Chapter Questions
2 MCQs