Unit Digit & Cyclicity

At a glance 11 formulas

1. 1Last digit onlyOnly the units digit of the base matters: $1287^{n}$ ends like $7^{n}$. In a sum or product, combine just the units digits.
2. 2Units-digit cycles

   0, 1, 5, 6

   itself, always

   4

   4, 6

   9

   9, 1

   2

   2, 4, 8, 6

   3

   3, 9, 7, 1

   7

   7, 9, 3, 1

   8

   8, 4, 2, 6

3. 3Exponent mod 4$a^{n} \text{ ends like } a^{r},\ r = n \bmod 4$$r = 0 \Rightarrow \text{ends like } a^{4}$
4. 4Quick remainder by 4Only the last two digits of the exponent decide its remainder by 4: for 2027, $27 = 24 + 3$ gives 3.
5. 5DifferencesIf the larger number has the smaller units digit, borrow 10: a number ending in 4 minus one ending in 5 ends in $14 - 5 = 9$.
6. 6Products with a 5A product with a factor 5 and any even factor ends in 0; a 5 times only odd numbers ends in 5.
7. 7Sums of factorials$n! \text{ ends in } 0 \text{ for } n \ge 5$$1! + 2! + 3! + 4! = 33$
8. 8Remainder by 10The remainder by 10 is the units digit; by 5 or by 2, it is that digit's own remainder by 5 or 2.
9. 9Cycle sum is 20The four units digits in one cycle of 2, 3, 7 or 8 add to 20, so in $a + a^{2} + \cdots + a^{n}$ only the terms left after full fours count.
10. 10Power of a powerIn $a^{b^{c}}$ find $b^{c} \bmod 4$ first: $b \equiv 1$ gives 1, $b \equiv 3$ gives 3 for odd $c$ and 1 for even $c$, and even $b$ gives 0 once $c \ge 2$.
11. 11Last two digitsReach a power that ends in 01, then reduce: $7^{4} = 2401$ and $3^{20}$ ends in 01, so $7^{4k}$ and $3^{20k}$ end in 01.