Find the unit digit of 2^6 + 3^7 + 4^8 + 5^9.
Answer (1)
To find the unit digit of the expression 2 6 + 3 7 + 4 8 + 5 9 , let's proceed step-by-step by identifying patterns in the unit digits of powers of each number individually.
Step 1: Find the unit digit of 2 6 : The unit digits of powers of 2 follow a repeating cycle: 2, 4, 8, 6.
2 1 ends in 2
2 2 ends in 4
2 3 ends in 8
2 4 ends in 6
The cycle repeats every four powers.
For 2 6 , the exponent 6 is equivalent to 6 mod 4 = 2 . Therefore, the unit digit of 2 6 is the same as 2 2 , which is 4.
Step 2: Find the unit digit of 3 7 : The unit digits of powers of 3 follow a repeating cycle: 3, 9, 7, 1.
3 1 ends in 3
3 2 ends in 9
3 3 ends in 7
3 4 ends in 1
The cycle repeats every four powers.
For 3 7 , the exponent 7 is equivalent to 7 mod 4 = 3 . Therefore, the unit digit of 3 7 is the same as 3 3 , which is 7.
Step 3: Find the unit digit of 4 8 : The unit digits of powers of 4 follow a repeating cycle: 4, 6.
4 1 ends in 4
4 2 ends in 6
The cycle repeats every two powers.
For 4 8 , the exponent 8 is equivalent to 8 mod 2 = 0 , which means it follows the cycle end result of 6.
Step 4: Find the unit digit of 5 9 : The unit digits of powers of 5 always end in 5 regardless of the exponent. Therefore, the unit digit of 5 9 is always 5.
Step 5: Sum the unit digits: Now, add the unit digits you have found:
4 + 7 + 6 + 5 = 22
The unit digit of 22 is 2. Therefore, the unit digit of 2 6 + 3 7 + 4 8 + 5 9 is 2.
