Unit Digit of Powers: The Cyclicity Method Step by Step

When you multiply two numbers, the unit digit of the answer depends only on the unit digits of the numbers you multiplied. That single fact makes these questions easy. To find the last digit of 243¹⁷, you never need 243 itself; you only need 3¹⁷, and in fact only the last digit of that.

Keep multiplying a digit by itself and the unit digits soon start repeating. For 2 the sequence of unit digits is 2, 4, 8, 6, 2, 4, 8, 6 and so on. The repeating block is called the cycle, and its length is the cyclicity. Once you know the cycle, any power is a matter of finding the right position in it.

Two memory aids help. The cycles for 2 and 8 contain the same four digits, as do the cycles for 3 and 7. And 4 and 9 alternate: an odd power of 4 ends in 4 and an even power ends in 6, while an odd power of 9 ends in 9 and an even power ends in 1.

1. Keep only the unit digit of the base.
2. Divide the exponent by 4 and note the remainder.
3. A remainder of 1, 2 or 3 picks the first, second or third digit of the cycle. A remainder of 0 picks the fourth.

For bases ending in 4 or 9, just check whether the exponent is odd or even. 4³⁵ has an odd exponent, so its unit digit is 4.

Likewise 9⁴⁸ has an even exponent, so its unit digit is 1.

For a product of powers, find each unit digit separately and multiply them, keeping only the last digit. For a sum, add the unit digits. For a difference, subtract them, borrowing 10 if the first digit is smaller, provided the first number is the larger one.

Questions sometimes give a huge exponent written as an expression, such as 7^(4k + 3). You do not need the value of k: the remainder on division by 4 is 3, so the unit digit is the third in the cycle of 7, which is 3.

The same thinking settles questions about factorials. From 5! onwards every factorial ends in 0, so in a sum such as 1! + 2! + … + 50! only the first four terms affect the unit digit: 1 + 2 + 6 + 24 = 33, giving a unit digit of 3.

• ✘ Taking the first digit of the cycle when the remainder is 0. Remainder 0 means the fourth (last) digit.
• ✘ Working with the whole base. Only the unit digit of the base matters.
• ✘ Adding exponents when multiplying different bases. Find each unit digit separately, then multiply.
• ✘ Forgetting to borrow in a difference. 2 − 7 becomes 12 − 7 = 5.

Cyclicity also drives remainder questions on large powers, and it pairs with divisibility rules in number-system sets. The core formulas are listed in the Quant cheat code.

Why do I divide the exponent by 4 and not by the length of each cycle?

Every digit's cycle has length 1, 2 or 4, and each of those divides 4. Using 4 for every digit therefore always lands on the right place in the cycle, so you only need to remember one rule.

What if the remainder is zero?

A remainder of zero means the exponent is a whole number of complete cycles, so take the last digit of the cycle, not the first. For 2 that is 6, for 3 it is 1, for 7 it is 1 and for 8 it is 6.

All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.

• Quant Cheat Code: 20 Must-Know Shortcuts & Formulas