Unit Digit Concepts
The digit at the end of a number is known as its unit digit. For instance, eight is the unit digit of 348 and one is the unit digit of 5721. Questions about unit digits belong to the field of Number System and Quantitative Aptitude. When handling questions of this type—particularly those involving very high powers—it is not necessary to work out the complete value. Rather, the problem can be solved by concentrating on the digit and noting the pattern that is formed by its successive powers.
Unit Digit Cycles
For example, consider the powers of 2. We have:
| Power | Value | Unit Digit |
|---|---|---|
| 2¹ | 2 | 2 |
| 2² | 4 | 4 |
| 2³ | 8 | 8 |
| 2⁴ | 16 | 6 |
Now 2 to the power of 5 equals 32, so the unit digit is 2; the pattern therefore begins to repeat with 2, 4, 8, 6 and then returns to 2. This repeated sequence is known as the cycle of the unit digits. The same principle holds for the other digits. For example, the unit digits of powers of 3 are 3, 9, 7, 1 and then the pattern repeats itself every four powers. In a similar way, the powers of 7 have the digits 7, 9, 3, 1. The powers of 8 give the digits 8, 4, 2, 6.
The digits 4 and 9 have two-digit cycles; the powers of 4 alternate between 4 and 6, and the powers of 9 alternate between 9 and 1. We can summarize the cycle patterns as follows:
| Last Digit of Base | Unit Digit Cycle | Cycle Length |
|---|---|---|
| 0 | 0 | 1 |
| 1 | 1 | 1 |
| 2 | 2, 4, 8, 6 | 4 |
| 3 | 3, 9, 7, 1 | 4 |
| 4 | 4, 6 | 2 |
| 5 | 5 | 1 |
| 6 | 6 | 1 |
| 7 | 7, 9, 3, 1 | 4 |
| 8 | 8, 4, 2, 6 | 4 |
| 9 | 9, 1 | 2 |
Finding the Unit Digit
An important rule is that the final digit of the base is significant. For example, if we are trying to find the unit digit of 23²⁰²⁵, there is no need to use the full number 23; since the last digit of 23 is 3, we can consider it as 3²⁰²⁵ when calculating the unit digit. In the same way, 684³⁵ becomes 4³⁵ and 127⁸⁵ becomes 7⁸⁵. When we have determined the cycle, we divide the exponent by the length of the cycle and the remainder shows us which position in the cycle to choose.
For example, let’s consider 7²⁰²⁶. The cycle of 7 is 7, 9, 3, 1 and its length is 4. Since 2026 divided by 4 gives a remainder of 2, we take the second digit in the cycle, which is 9. Hence, the unit digit of 7²⁰²⁶ is 9.
With regard to a remainder of zero, note the example of 2²⁰²⁴. The cycle of 2 is 2, 4, 8, 6. As 2024 is divisible by 4, the remainder is 0; in this instance we use the last number in the cycle rather than the first. Hence the unit digit of 2²⁰²⁴ is 6. When the remainder is 1 it corresponds to the first number, 2 to the second, 3 to the third, and 0 to the fourth.
Special Cases: Bases Ending in 4 and 9
There is a method that applies to numbers ending in 4 or 9. The powers of 4 alternate between 4 and 6; therefore, if the exponent is odd the unit digit will be 4 but if it is even the unit digit will be 6. For instance, 4²⁰²⁵ has a unit digit of 4 since 2025 is odd. With regard to 9, the powers alternate between 9 and 1; 9²⁰²⁶ has a unit digit of 1 because 2026 is even.
Digits such as 0, 1, 5 and 6 are simpler: all positive powers of 0 end in 0, all powers of 1 end in 1, all positive powers of 5 end in 5 and all positive powers of 6 end in 6; therefore, if the last digit of the base is one of these, there is no need for cycle calculations.
Unit Digit in Addition, Subtraction and Multiplication
Unit digit problems may also include addition, subtraction or multiplication; in these instances, first determine the unit digit of each term individually and then carry out the operation using those unit digits.
For example, consider 7¹⁰⁰ and 3¹⁰¹. The unit digit of 7¹⁰⁰ is 1 since division of 100 by 4 gives a remainder of 0. The unit digit of 3¹⁰¹ is 3 as 101 has a remainder of 1. When adding them:
1 + 3 = 4
Thus, the final unit digit is 4. The method also applies to multiplication. Consider 13¹⁵⁷ × 17⁸⁴. The last digits are 3 and 7. The unit digit of 3¹⁵⁷ is 3 (as 157 divided by 4 leaves a remainder of 1). The unit digit of 7⁸⁴ is found by dividing 84 by 4, which gives 21 with no remainder; therefore, we use the last digit in the cycle of 7, which is 1. When we multiply the unit digits together:
3 × 1 = 3
Hence, the final unit digit is 3.
Large Bases
Large bases don't make the problem any more difficult. Take 6792¹⁰⁰⁰ as an example. The digit in the units place is important—here it is 2. The cycle for 2 is 2, 4, 8, 6. Since 1000 is divisible by 4, the remainder is 0. We then take the last digit in that cycle: 6. Hence, the units digit of 6792¹⁰⁰⁰ is 6.
A useful point is this: if both the factors 2 and 5 are present in a product, the result will end with 0 since 2 multiplied by 5 is 10. For instance, 2⁵ times 5⁵ equals 10⁵, a number which obviously ends in 0. In mathematical expressions, identifying such pairs helps to save time.
Expressions with Exponents Outside the Brackets
Advanced questions can involve exponents that are outside the brackets. For example, in (9²⁰²⁵ + 8²⁰²⁶)²⁰²⁷ we should not begin with the exponent. Instead, we first simplify the expression inside the brackets to find its unit digit. After having determined the unit digit of the bracket, we use that digit as the base for the power. Thus, the calculations are carried out from inside to outside.
There are cases in which the exponent is itself a power. Take 7^(3¹⁰⁰) for example.
In order to determine the unit digit, there is no need to work out 3¹⁰⁰. Since the cycle for 7 is 4, all we need to do is find the remainder obtained when 3¹⁰⁰ is divided by 4. Now that 3² equals 9, which has a remainder of 1 when divided by 4, it follows that all even powers of 3 have a remainder of 1. Hence, 3¹⁰⁰ has a remainder of 1.
We must then select the number in the cycle of 7: 7. Therefore, the unit digit is 7.
The key concept in unit digit problems is straightforward after you've worked out the repeating patterns; rather than calculating the large numbers, reduce the base to its final digit, find out which cycle it fits into, and use the exponent to locate the correct position within that cycle, after which you can carry out any addition, subtraction or multiplication using the unit digits.
Important Cycles for Exams
For exams, the most important cycles to remember are:
| Base | Unit Digit Cycle |
|---|---|
| 2 | 2, 4, 8, 6 |
| 3 | 3, 9, 7, 1 |
| 4 | 4, 6 |
| 7 | 7, 9, 3, 1 |
| 8 | 8, 4, 2, 6 |
| 9 | 9, 1 |
When these patterns are known, expressions involving large numbers and big exponents can be worked out quickly without the need for lengthy calculations.
The fundamental method can be summed up in a single statement: examine the last digit of the base, determine its cycle, divide the exponent by the length of the cycle, and then use the remainder to select the appropriate unit digit.
Practice Questions
Directions (1–50): Find the unit digit of each of the following numbers.
Answer Key
| Q | Answer | Q | Answer |
|---|---|---|---|
| Q1 | D) 9 | Q26 | D) 9 |
| Q2 | B) 3 | Q27 | E) None of These |
| Q3 | B) 3 | Q28 | E) None of These |
| Q4 | E) None of These | Q29 | D) 8 |
| Q5 | E) None of These | Q30 | E) None of These |
| Q6 | A) 0 | Q31 | D) 7 |
| Q7 | B) 3 | Q32 | E) None of These |
| Q8 | E) None of These | Q33 | E) None of These |
| Q9 | D) 6 | Q34 | E) None of These |
| Q10 | A) 2 | Q35 | D) 8 |
| Q11 | D) 8 | Q36 | E) None of These |
| Q12 | E) None of These | Q37 | E) None of These |
| Q13 | B) 3 | Q38 | A) 0 |
| Q14 | D) 8 | Q39 | C) 4 |
| Q15 | D) 8 | Q40 | A) 0 |
| Q16 | E) None of These | Q41 | B) 3 |
| Q17 | A) 2 | Q42 | C) 7 |
| Q18 | E) None of These | Q43 | D) 8 |
| Q19 | C) 7 | Q44 | C) 4 |
| Q20 | E) None of These | Q45 | C) 7 |
| Q21 | A) 0 | Q46 | E) None of These |
| Q22 | E) None of These | Q47 | B) 4 |
| Q23 | B) 2 | Q48 | E) None of These |
| Q24 | B) 3 | Q49 | B) 3 |
| Q25 | E) None of These | Q50 | A) 0 |