Number System

high importance~2 Q in Tier 133 formulas⚡ 18 shortcuts6 subtopics

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Subtopic 4 of 6·← Remainders & remainder theorem·Factors, prime factorisation & trailing zeros →

Unit digit & cyclicity

The unit digit of a power depends only on the unit digit of the base and the exponent's position in the cycle.

Unit digit of base	Cycle	Length
0, 1, 5, 6	same digit always	1
4	4, 6	2 (odd power → 4, even → 6)
9	9, 1	2 (odd power → 9, even → 1)
2	2, 4, 8, 6	4
3	3, 9, 7, 1	4
7	7, 9, 3, 1	4
8	8, 4, 2, 6	4

Method: divide the exponent by 4 (only its last two digits matter). Remainder r → use power r; remainder 0 → use power 4.

For products, multiply only the unit digits. Any even number × any number ending in 5 ends in 0. Every n! with n ≥ 5 ends in 0.

📖 Notes 🧩 5 question types ⚡ Formulas 🎯 13 practice Q

Detailed notes

What is the unit digit?

The unit digit is the last digit of a number (the ones place). In 4,587 the unit digit is 7. Exams ask for the unit digit of huge expressions like $7^{95} \times 3^{58}$ — numbers far too big to calculate. The good news: the unit digit of a product depends only on the unit digits of the numbers being multiplied. $1237 \times 48$ ends in the same digit as $7 \times 8 = 56$ , i.e. 6.

Powers repeat in a cycle

Write the powers of 7: $7, 49, 343, 2401, 16807 \dots$ Unit digits: 7, 9, 3, 1, 7, 9, 3, 1 … The pattern repeats every 4 powers. This repeating length is called cyclicity.

Unit digit of base	Cycle of unit digits	Cycle length
0, 1, 5, 6	stays the same	1
4	4, 6	2 (odd power 4, even power 6)
9	9, 1	2 (odd power 9, even power 1)
2	2, 4, 8, 6	4
3	3, 9, 7, 1	4
7	7, 9, 3, 1	4
8	8, 4, 2, 6	4

The method for any power

Keep only the unit digit of the base: $2137^{754}$ → base digit 7.
Divide the exponent by 4 (only the last two digits of the exponent matter because 100 is a multiple of 4).
Remainder 1, 2, 3 → use that power. Remainder 0 → use power 4.
Read the unit digit: $754 \bmod 4 = 2$ → $7^2 = 49$ → answer 9.

Products, sums and differences

Product: find each unit digit, multiply them, keep the last digit.
Sum: add the unit digits, keep the last digit.
Difference: subtract the unit digits; if the result is negative, add 10. Example: $7^{105} - 3^{58}$ → 7 and 9 → $7 - 9 = -2$ → $10 - 2 = 8$ . (This works only when the first number is actually bigger.)

Shortcuts that save time

Any even number × any number ending in 5 → unit digit 0.
An odd number × a number ending in 5 → unit digit 5 (so $1 \times 3 \times 5 \times \dots \times 99$ ends in 5).
$n!$ ends in 0 for every $n \ge 5$ . So in $1! + 2! + \dots + 50!$ only $1! + 2! + 3! + 4! = 33$ matters → unit digit 3.
Product of numbers ending in 1 and 6 keep the same digit when raised to any power.

Power towers

For $17^{17^{17}}$ you need the exponent $17^{17}$ divided by 4. Since 17 leaves 1 with 4, $17^{17}$ also leaves 1. So the answer is the unit digit of $7^1 = 7$ . For $3^{4^5}$ the exponent is a multiple of 4, so use $3^4$ → 1.

Last two digits (bonus)

Numbers ending in 1: last two digits of $(\dots a1)^n$ are: tens digit $= a \times (\text{unit digit of } n)$ (keep its last digit), units digit 1. Example: $31^{786}$ → $3 \times 6 = 18$ → 8 → 81.
Look for a power ending in 01: $7^4 = 2401$ , so $7^{4k}$ ends in 01.

Quick revision

Only the base's unit digit and the exponent mod 4 matter.
Remainder 0 → use the 4th power.
0, 1, 5, 6 never change; 4 and 9 have cycle 2.
Even × 5 → 0; odd × 5 → 5; $n! \ (n \ge 5)$ → 0.
Negative difference of unit digits → add 10.

Types of questions asked

Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.

Type 1: Unit digit of a power or a product of powersvery common3 practice Q

How to spot it:

'Find the unit digit of $2137^{754}$ ' or of $4^{63} \times 9^{72} \times 8^{41}$ .

\text{unit}(a^n) = \text{unit}\big(a^{\,n \bmod 4}\big),\ \text{use } 4 \text{ if } n \bmod 4 = 0

Keep only the unit digit of each base.
Reduce each exponent mod 4 (mod 2 is enough for 4 and 9).
Find each unit digit and multiply them; keep the last digit.

Why: unit digits of powers repeat every 4 steps (or fewer).

Example: Find the unit digit of $(1573)^{47}$ .

Base digit 3, $47 \bmod 4 = 3$ → $3^3 = 27$ → unit digit 7.

Type 2: Unit digit of a sum or difference of powerscommon2 practice Q

How to spot it:

Powers joined by + or −, e.g. $7^{105} - 3^{58}$ or $2^{51} + 3^{52} + 4^{53} + 5^{54}$ .

Find the unit digit of each power separately.
Add (or subtract) those digits.
Keep the last digit; if a difference is negative, add 10.

Why: carrying and borrowing never affect the ones place beyond what the unit digits show.

Example: Find the unit digit of $6^{25} + 9^{31}$ .

$6^{25}$ → 6. $9^{31}$ (odd) → 9. $6 + 9 = 15$ → unit digit 5.

Type 3: Unit digit of long products and factorial sumscommon3 practice Q

How to spot it:

A product of several plain numbers, a product like $1 \times 3 \times 5 \times \dots$ , or $1! + 2! + \dots + n!$ .

Look for an even number together with a 5 → answer 0 immediately.
For factorial sums, drop every term from $5!$ on (they end in 0).
Otherwise multiply unit digits step by step, keeping only the last digit.

Why: 2 × 5 = 10 produces a zero that stays forever.

Example: Find the unit digit of $1! + 2! + 3! + \dots + 99!$ .

Terms from $5!$ end in 0. $1 + 2 + 6 + 24 = 33$ → unit digit 3.

Type 4: Power towers (exponent is itself a power)occasional2 practice Q

How to spot it:

Expressions like $17^{17^{17}}$ or $3^{4^5}$ .

a^{b^c}:\ \text{find } b^c \bmod 4 \text{ first}

Find the remainder of the exponent $b^c$ when divided by 4 (use: odd $b$ with $b \bmod 4 = 1$ → 1; $b \bmod 4 = 3$ → 3 for odd $c$ , 1 for even $c$ ; multiple of 4 → 0).
Use that remainder on the cycle of the base.

Why: only the exponent's position in the 4-cycle matters.

Example: Unit digit of $3^{4^5}$ ?

$4^5$ is a multiple of 4 → remainder 0 → use $3^4 = 81$ → 1.

Type 5: Last two digits of a poweroccasional2 practice Q

How to spot it:

'Find the last two digits of $31^{786}$ ' or of $7^{2008}$ .

(10a+1)^n \text{ ends in } \big[(a \cdot n) \bmod 10\big]1

If the base ends in 1: tens digit = (tens digit of base × unit digit of exponent), last digit only; units digit 1.
Otherwise find a power that ends in 01 (e.g. $7^4 = 2401$ , $3^{20}$ ) and reduce the exponent.

Why: $(10a + 1)^n = 1 + 10an + (\text{multiples of } 100)$ .

Example: Last two digits of $41^{27}$ ?

$4 \times 7 = 28$ → 8. Last two digits 81.

Formulas

Cyclicity rule

\text{u.d.}(a^n) = \text{u.d.}(a^{r}),\ r = n \bmod 4,\ (r = 0 \Rightarrow r = 4)

Cycles of 2, 3, 7, 8

2:\,2,4,8,6 \quad 3:\,3,9,7,1 \quad 7:\,7,9,3,1 \quad 8:\,8,4,2,6

Factorials

n! \equiv 0 \pmod{10} \text{ for } n \ge 5

Shortcut tricks

⚡ Last two digits of the exponent

For mod 4, only the last two digits of the exponent matter (100 is a multiple of 4).

Example: Find the unit digit of $1357^{2463}$ .

Base ends in 7. $63 \bmod 4 = 3$ ⇒ $7^3 = 343$ ⇒ unit digit 3.

⚡ Even × 5 = 0

If the product contains an even factor and a factor ending in 5, the unit digit is 0 — no cycle work needed.

Example: Find the unit digit of $2^{31} \times 5^{17} \times 3^{9}$ .

$2^{31}$ is even and $5^{17}$ ends in 5 ⇒ product ends in 0.

⚡ Factorial sums: only the first four terms

From 5! onwards every factorial ends in 0, so the unit digit of 1! + 2! + … + n! (n ≥ 4) is always 3.

Example: Find the unit digit of 1! + 2! + 3! + … + 50!.

1 + 2 + 6 + 24 = 33; all later terms end in 0 ⇒ unit digit 3.

Where students lose marks

Treating exponent remainder 0 as power 0 (unit digit 1) — use power 4 instead.
Taking the exponent mod 4 using only its last digit (use the last two digits).
For a difference like $7^{a} - 3^{b}$ , forgetting to borrow 10 when the first unit digit is smaller.

Practice sets — 13 questions

Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.

Topic test · 13 questions

Suggested time 9 min · wrong answers go to your mistake notebook automatically.