Unit Digits and Cyclic Patterns

Learning outcome

Find unit digits of large powers using short cycles, including exponents whose cycle remainder is zero.

Concepts and reasons

The unit digit is the last digit of a nonnegative integer written in base ten. Here bases are nonnegative integers and exponents are positive integers. The notation a^n means a product of n copies of a; for n = 1, it is simply a.

Only unit digits matter when finding a product’s unit digit. If two numbers are 10A + a and 10B + b, their product differs from a × b by a multiple of 10. Similarly, a sum’s unit digit depends only on the sum of its unit digits.

Repeated multiplication produces these cycles, listed from exponent 1:

Digits 0, 1, 5 and 6 keep their own digit for every positive exponent.

Digit 4 follows 4, 6; digit 9 follows 9, 1.

Digit 2 follows 2, 4, 8, 6; digit 3 follows 3, 9, 7, 1.

Digit 7 follows 7, 9, 3, 1; digit 8 follows 8, 4, 2, 6.

The next unit digit depends only on the current unit digit and the base’s last digit. Once a digit repeats, the same steps follow again; this explains the cycles.

Divide the exponent by the cycle length. A positive remainder selects that position. Remainder 0 selects the last position, not digit 0, because complete cycles end there.

Worked examples

Example 1. Find the unit digit of 7^23. The cycle is 7, 9, 3, 1. Since 23 = 4 × 5 + 3, select the third entry: 3.

Example 2. Find the unit digit of 12^20. Use the cycle for 2: 2, 4, 8, 6. Since 20 divides exactly by 4, select the fourth entry: 6.

Example 3. Find the unit digit of 34^7 × 19^8. The odd power of a number ending in 4 ends in 4. The even power of a number ending in 9 ends in 1. Thus the product ends in 4 × 1 = 4.

Common traps

Do not multiply the base by the exponent. Start cycles at exponent 1. Use the exponent’s remainder, not merely its final digit. After adding or multiplying final digits, keep only the result’s unit digit.

Practice questions

1. Find the unit digit of 3^14.
2. Find the unit digit of 28^16.
3. Find the unit digit of 17^5 + 24^6.

Answers and explanations

1. The cycle for 3 is 3, 9, 7, 1. Since 14 leaves remainder 2 upon division by 4, the second entry gives 9.
2. The cycle for 8 has length 4. Since 16 leaves remainder 0 upon division by 4, select the fourth digit, 6.
3. The first power ends in 7 because 5 leaves remainder 1 upon division by 4. The second ends in 6 because its exponent is even. Their sum ends in the unit digit of 7 + 6 = 13, which is 3.