Each of the aptitude question exam papers contains 2 or more questions on identifying unit place digit for some number(x) to the power of some other number (n).
Let us look at 1 of the example: What will be the unit’s place digit for 77^123? -> Do you really feel that we are supposed to multiply 77 number 123 times during the exam? -> Obviously Not! We need to work with the way so that without doing that much multiplication we should be capable of answering. The power cycle will help us to solve this problem.
Let us take a real-life example to understand the power cycle. Let us assume Sun as a number for which index n is to be calculated. Sun follows the below 2 steps: Step 1: Sunrise Step 2: Sunset So let us look at power and respective values for Sun to the power n. Sun^1 -> Sunrise Sun^2 -> Sunset Sun^3 -> Sunrise Sun^4 -> Sunset Sun^5 -> Sunrise Looking at the above table we can see that Sun follows the pattern of Sunrise, Sunset. (Sunrise, Sunset) is the Power cycle for Sun.
We will identify power cycles for all numbers 0 to 9 and you would be able to solve any problem asked in the exam easily and post-practice orally as well. If you can remember power cycle values that would be great however even if you are unable to remember them, we will look at a technique to calculate the power cycle of numbers in less than 1 minute during the exam as well and you can solve the problem. Power Cycles for all Numbers (Focus on Unit place digit only)
Number 0:
Let us calculate the values for 0^1 to 0^5: 0^1: 0 0^2: 0 0^3: 0 0^4: 0 0^5: 0 So as you can observe: The value of 0^1 to 0^5 is 0 only. So the unit place digit is 0 for any power of 0. Power cycle for 0: (0)
Question 1:
What will be unit’s place digit for 250^123?
Solution:
Look at the unit place of number 250. The unit’s place digit is 0. Power Cycle of 0: (0)
The answer is unit’s place for 250^123 will be 0.
Question 2:
What will be unit’s place digit for 670^4123?
Solution:
Look at the unit place of number 670. The unit’s place digit is 0. Power Cycle of 0: (0)
The answer is unit’s place for 670^4123 will be 0.
Number 1:
Let us calculate the values for 1^1 to 1^5: 1^1: 1 1^2: 1 1^3: 1 1^4: 1 1^5: 1 So as you can observe: The value of 1^1 to 1^5 is 1 only. So the unit place digit is 1 for any power of 1. Power cycle for 1: (1)
Question 1:
What will be unit’s place digit for 121^53?
Solution:
Look at the unit place of number 121. The unit’s place digit is 1. Power Cycle of 1: (1)
The answer is unit’s place digit for 121^53 will be 1.
Question 2:
What will be unit’s place digit for 791^5643?
Solution:
Look at the unit place of number 791. The unit’s place digit is 1. Power Cycle of 1: (1)
The answer is unit’s place digit for 791^5643 will be 1.
Let us calculate the values for 2^1 to 2^5: 2^1: 2 2^2: 4 2^3: 8 2^4: 16 2^5: 32 So as you can observe: The unit place digit of 2^1 to 2^5 is in order 2, 4, 8, 6 and it will keep repeating as 2, 4, 8, 6 Power cycle for 2: (2, 4, 8, 6)
Question 1:
What will be unit’s place digit for 2^33?
Solution:
Look at the unit place of number 2. Unit’s place digit is 2. Power Cycle of 2 : (2, 4, 8, 6) There are total 4 values which keep repeating always for power of 2. Now look at index which is to be identified: 33 As 4 numbers keep on repeating for power cycle of 2, we need to divide 33 by 4 and identify remainder of it so that we can understand what can be unit place number. (Index to be found / Size of power cycle) = 33/4, Quotient = 8 and Remainder = 1. 1 -> 2 2 -> 4 3 -> 8 0 -> 6 You don’t need to remember this table you just need to make sure as you know pattern of power cycle you have to reach till index number. Like in this case: To reach 33 and you have size of 4 4, 8, 12…..32 so 32nd index would be last number in power cycle that is 6 33rd index would have 1st number in power cycle that is 2.
The answer is unit’s place digit for 2^33 will be 2.
Question 2:
What will be unit’s place digit for 1222^438?
Solution:
Look at the unit place of number 1222. Unit’s place digit is 2. Power Cycle of 2 : (2, 4, 8, 6) There are total 4 values which keep repeating always for power of 2. Now look at index which is to be identified: 438 As 4 numbers keep on repeating for power cycle of 2, we need to divide 438 by 4 and identify remainder of it so that we can understand what can be unit place number. (Index to be found / Size of power cycle) = 438/4, Quotient = 109 and Remainder = 2. 1 -> 2 2 -> 4 3 -> 8 0 -> 6
The answer is unit’s place digit for 1222^438 will be 4.
Let us calculate the values for 3^1 to 3^5: 3^1: 3 3^2: 9 3^3: 27 3^4: 81 3^5: 243 So as you can observe: The unit place digit of 3^1 to 3^5 is in order 3, 9, 7, 1 and it will keep repeating as 3, 9, 7, 1 Power cycle for 3: (3, 9, 7, 1)
Question 1:
What will be unit’s place digit for 3^36?
Solution:
Look at the unit place of number 3. Unit’s place digit is 3. Power Cycle of 3 : (3, 9, 7, 1) There are total 4 values which keep repeating always for power of 2. Now look at index which is to be identified: 36 As 4 numbers keep on repeating for power cycle of 3, we need to divide 36 by 4 and identify remainder of it so that we can understand what can be unit place number. (Index to be found / Size of power cycle) = 36/4, Quotient = 9 and Remainder = 0. 1 -> 3 2 -> 9 3 -> 7 0 -> 1
The answer is unit’s place digit for 3^36 will be 1.
Question 2:
What will be unit’s place digit for 123^498?
Solution:
Look at the unit place of number 123. Unit’s place digit is 3. Power Cycle of 3 : (3, 9, 7, 1) There are total 4 values that keep repeating always for a power of 3. Now look at index which is to be identified: 498 As 4 numbers keep on repeating for power cycle of 3, we need to divide 498 by 4 and identify remainder of it so that we can understand what can be unit place number. (Index to be found / Size of power cycle) = 498/4, Quotient = 124 and Remainder = 2. 1 -> 3 2 -> 9 3 -> 7 0 -> 1
The answer is unit’s place digit for 123^498 will be 9.
Number 4:
Let us calculate the values for 4^1 to 4^5: 4^1: 4 4^2: 16 4^3: 64 4^4: 256 4^5: 1024 So as you can observe: The unit place digit of 4^1 to 4^5 is in order 4, 6 and it will keep repeating as 4, 6 Power cycle for 4: (4, 6)
Question 1:
What will be unit’s place digit for 4^360?
Solution:
Look at the unit place of number 4. Unit’s place digit is 4. Power Cycle of 4 : (4, 6) There are total 2 values which keep repeating always for power of 4. Now look at index which is to be identified: 360 As 2 numbers keep on repeating for power cycle of 4, we need to divide 360 by 2 and identify remainder of it so that we can understand what can be unit place number. (Index to be found / Size of power cycle) = 360/2, Quotient = 180 and Remainder = 0. 1 -> 4 0 -> 6
The answer is unit’s place digit for 4^360 will be 6.
Question 2:
What will be unit’s place digit for 1234^6987?
Solution:
Look at the unit place of number 1234. Unit’s place digit is 4. Power Cycle of 4: (4, 6) There are total 2 values which keep repeating always for power of 4. Now look at index which is to be identified: 6987 As 2 numbers keep on repeating for power cycle of 4, we need to divide 6987 by 2 and identify remainder of it so that we can understand what can be unit place number. (Index to be found / Size of power cycle) = 6987/2, Quotient = 3493 and Remainder = 1. 1 -> 4 0 -> 6
The answer is unit’s place digit for 1234^6987 will be 4.
Let us calculate the values for 5^1 to 5^5: 5^1: 5 5^2: 25 5^3: 125 5^4: 625 5^5: 3125 So as you can observe: The unit place digit of 5^1 to 5^5 is 5 only. So, unit place digit of 5 for any power 5. Power cycle for 5: (5)
Question 1:
What will be unit’s place digit for 5^12?
Solution:
Look at the unit place of number 5. Unit’s place digit is 5. Power Cycle of 5 : (5)
The answer is unit’s place digit for 5^12 will be 5.
Question 2:
What will be unit’s place digit for 25^56?
Solution:
Look at the unit place of number 25. Unit’s place digit is 5. Power Cycle of 5 : (5)
The answer is unit’s place digit for 25^56 will be 5.
Number 6:
Let us calculate the values for 6^1 to 6^5: 6^1: 6 6^2: 36 6^3: 216 6^4: 1296 6^5: 7776 So as you can observe: The unit place digit of 6^1 to 6^5 is 6 only. So, Unit place digit is 6 for any power of 6. Power cycle for 6: (6)
Question 1:
What will be unit’s place digit for 56^142?
Solution:
Look at the unit place of number 56. Unit’s place digit is 6. Power Cycle of 6 : (6)
The answer is unit’s place digit for 56^142 will be 6.
Question 2:
What will be unit’s place digit for 286^56?
Solution:
Look at the unit place of number 286. Unit’s place digit is 6. Power Cycle of 6 : (6)
The answer is unit’s place digit for 286^56 will be 6.
Number 7:
Let us calculate the values for 7^1 to 7^5: 7^1: 7 7^2: 49 7^3: 343 7^4: 2401 7^5: 16807 So as you can observe: The unit place digit of 7^1 to 7^5 is in order 7, 9, 3, 1 and it will keep repeating as 7, 9, 3, 1 Power cycle for 7: (7, 9, 3, 1)
Question 1:
What will be unit’s place digit for 7^77?
Solution:
Look at the unit place of number 7. Unit’s place digit is 7. Power Cycle of 7 : (7, 9, 3, 1) There are total 4 values that keep repeating always for power of 7. Now look at index which is to be identified: 77 As 4 numbers keep on repeating for power cycle of 7, we need to divide 77 by 4 and identify remainder of it so that we can understand what can be unit place number. (Index to be found / Size of power cycle) = 77/4, Quotient = 19 and Remainder = 1. 1 -> 7 2 -> 9 3 -> 3 0 -> 1
The answer is unit’s place digit for 7^77 will be 7.
Question 2:
What will be unit’s place digit for 1237^496?
Solution:
Look at the unit place of number 1237. Unit’s place digit is 7. Power Cycle of 7 : (7, 9, 3, 1) There are total 4 values which keep repeating always for power of 7. Now look at index which is to be identified: 496 As 4 numbers keep on repeating for power cycle of 7, we need to divide 496 by 4 and identify remainder of it so that we can understand what can be unit place number. (Index to be found / Size of power cycle) = 496/4, Quotient = 124 and Remainder = 0. 1 -> 7 2 -> 9 3 -> 3 0 -> 1
The answer is unit’s place digit for 1237^496 will be 1.
Let us calculate the values for 8^1 to 8^5: 8^1: 8 8^2: 64 8^3: 512 8^4: 4096 8^5: 32768 So as you can observe: The unit place digit of 8^1 to 8^5 is in order 8, 4, 2, 6 and it will keep repeating as 8, 4, 2, 6 Power cycle for 8: (8, 4, 2, 6)
Question 1:
What will be unit’s place digit for 8^67?
Solution:
Look at the unit place of number 8. Unit’s place digit is 8. Power Cycle of 8 : (8, 4, 2, 6) There are total 4 values which keep repeating always for power of 8. Now look at index which is to be identified: 67 As 4 numbers keep on repeating for power cycle of 8, we need to divide 67 by 4 and identify remainder of it so that we can understand what can be unit place number. (Index to be found / Size of power cycle) = 67/4, Quotient = 14 and Remainder = 3. 1 -> 8 2 -> 4 3 -> 2 0 -> 6
The answer is unit’s place digit for 8^67 will be 2.
Question 2:
What will be unit’s place digit for 128^6802?
Solution:
Look at the unit place of number 128. Unit’s place digit is 8. Power Cycle of 8 : (8, 4, 2, 6) There are total 4 values which keep repeating always for power of 8. Now look at index which is to be identified: 6802 As 4 numbers keep on repeating for power cycle of 8, we need to divide 6802 by 4 and identify remainder of it so that we can understand what can be unit place number. (Index to be found / Size of power cycle) = 6802/4, Quotient = 1700 and Remainder = 2. 1 -> 8 2 -> 4 3 -> 2 0 -> 6
The answer is unit’s place digit for 128^6802 will be 4.
Number 9:
Let us calculate the values for 9^1 to 9^5: 9^1: 9 9^2: 81 9^3: 729 9^4: 6651 9^5: 59859 So as you can observe: The unit place digit of 9^1 to 9^5 is in order 9, 1 and it will keep repeating as 9, 1 Power cycle for 9: (9, 1)
Question 1:
What will be unit’s place digit for 9^99?
Solution:
Look at the unit place of number 9. Unit’s place digit is 9. Power Cycle of 9 : (9, 1) There are total 2 values which keep repeating always for power of 9. Now look at index which is to be identified: 99 As 2 numbers keep on repeating for power cycle of 9, we need to divide 99 by 2 and identify remainder of it so that we can understand what can be unit place number. (Index to be found / Size of power cycle) = 99/2, Quotient = 49 and Remainder = 1. 1 -> 9 0 -> 1
The answer is unit’s place digit for 9^99 will be 9.
Question 2:
What will be unit’s place digit for 999^1000?
Solution:
Look at the unit place of number 999. Unit’s place digit is 9. Power Cycle of 9 : (9, 1) There are total 2 values which keep repeating always for power of 9. Now look at index which is to be identified: 1000 As 2 numbers keep on repeating for power cycle of 9, we need to divide 1000 by 2 and identify remainder of it so that we can understand what can be unit place number. (Index to be found / Size of power cycle) = 1000/2, Quotient = 500 and Remainder = 0. 1 -> 9 0 -> 1
The answer is unit’s place digit for 999^1000 will be 1.
