1
The last digit only depends on the last digit.
When you multiply, nothing to the left can reach the units column.
3 × 7 = 21
also ends in 1
So the whole calculation can be done with single digits. 7100 is an
85-digit monster, and none of it matters except the end.
The units digit also settles a remainder on division by 10 or by 5, because 10 and 5
divide everything to its left. 2⁷ = 128 ends in 8, so 2⁷ ÷ 10 leaves 8, and 2⁷ ÷ 5
leaves 3 (8 ÷ 5 = 1 r 3; indeed 128 = 5 × 25 + 3).
2
Build the cycle.
List powers until a units digit comes back round.
| Power | Value | Units digit |
|---|
| 71 | 7 | 7 |
|---|
| 72 | 49 | 9 |
|---|
| 73 | 343 | 3 |
|---|
| 74 | 2401 | 1 |
|---|
| 75 | 16807 | 7 — it repeats |
|---|
79
31
79
cycle length 4
Stop the moment the first digit reappears. Everything after that is the same list
again, forever.
3
Divide the exponent by the cycle length.
100 ÷ 4 = 25 remainder 0
exactly 25 full cycles
remainder 0 → the last in the cycle
not the first
units digit = 1
the 4th entry
7100 ends in 1
A remainder of zero means the end of the cycle
Reading it as position 0 and taking the first entry gives 7, which is wrong. A
clean division lands you on the last step, not before the first.
4
Every digit's cycle.
| Ends in | Cycle | Length |
|---|
| 0, 1, 5, 6 | itself, always | 1 |
|---|
| 4 | 4, 6 | 2 |
|---|
| 9 | 9, 1 | 2 |
|---|
| 2 | 2, 4, 8, 6 | 4 |
|---|
| 3 | 3, 9, 7, 1 | 4 |
|---|
| 7 | 7, 9, 3, 1 | 4 |
|---|
| 8 | 8, 4, 2, 6 | 4 |
|---|
Nothing has a cycle longer than 4. Four multiplications is the worst case.
Numbers ending in 0, 1, 5 or 6 never change — 6999 ends in 6, no
working needed at all.
5
One more, with a leftover.
The units digit of 345.
cycle of 3: 3, 9, 7, 1
length 4
45 ÷ 4 = 11 remainder 1
one past the full cycles
1st in the cycle = 3
the answer
Only the base's own last digit matters, so 1345 and 2345 also
end in 3.
This is the same machinery as the remainder-patterns lesson. There the cycle was
in the remainders; here it is in the last digit.
6
Variations: a cycle of two, a big base, and an exponent that is itself a power.
A cycle of length 2: 9⁵⁸ and 24⁴⁰.
9: 9, 1, 9, 1 … odd → 9, even → 1
divide by 2, not 4
58 is even → 9⁵⁸ ends in 1
remainder 0: the last entry
24 ends in 4: 4, 6 … 40 is even → 6
only the last digit of the base
What is different: for 4 and 9 we only need to know whether the exponent is odd or even.
A big base: the units digit of 2043³⁸.
2043 ends in 3: cycle 3, 9, 7, 1
throw away 204
38 ÷ 4 = 9 remainder 2
the second entry
units digit = 9
same as 3³⁸
What is different: nothing — and that is the point. A four-digit base is no harder than a one-digit one.
A power in the exponent: the units digit of 8 raised to the power 3⁵.
exponent = 3⁵ = 243
work the top first
243 ÷ 4: last two digits 43 = 4 × 10 + 3
remainder 3
8: 8, 4, 2, 6 → third entry = 2
the answer
What is different: the exponent is not a plain number, so we evaluate it before dividing. Reading it as 8³ × 5 or (8³)⁵ changes the question.
7
Sums, products and differences of powers.
Find each power's units digit on its own, then combine just those digits.
| Expression | Each part | Units digit |
|---|
| 243 + 743 | 43 r 3 → 8 and 3 | 8 + 3 = 11 → 1 |
|---|
| 322 × 815 | 22 r 2 → 9; 15 r 3 → 2 | 9 × 2 = 18 → 8 |
|---|
| 720 − 310 | 20 r 0 → 1; 10 r 2 → 9 | borrow: 11 − 9 = 2 |
|---|
The units digit of 21 + 22 + … + 242?
one full cycle: 2 + 4 + 8 + 6 = 20
ends in 0
42 terms = 10 full cycles + 2 extra
the cycles add nothing to the units
extras 241 + 242 → 2 + 4 = 6
remainders 1 and 2
Combine the units digits, then take the units digit again
8 + 3 = 11 ends in 1, not 11. For a difference, the bigger number may end in the smaller
digit — 720 ends in 1 and 310 in 9, so borrow: 11 − 9 = 2, not 9 − 1 = 8.
On the exam
Units-digit items use one power, or a sum, product or difference of two, and sometimes a
long sum of powers. Keep only the base's last digit, divide the exponent by 4 (use its last
two digits), and remember remainder 0 means the last entry of the cycle.
✓
Check: test the rule on a small case.
7⁴
2401
4 ÷ 4 = r0 → last = 1 ✓
7⁸
5,764,801
8 ÷ 4 = r0 → 1 ✓
3⁵
243
5 ÷ 4 = r1 → 3 ✓
Remember the flow
Last digit only · Build the cycle · Divide the exponent
Remainder 0 means the last in the cycle.
