Numbers are the grammar of quant — from counting to clock arithmetic. Master definitions, divisibility shortcuts, prime-factorization formulas, factorial zeros, and remainder theorems (Fermat, Wilson, Euler) — every later module reuses them.
| Set | Definition | Examples |
|---|---|---|
| Natural Numbers | Counting from 1 | 1, 2, 3, 4… |
| Whole Numbers | Naturals + 0 | 0, 1, 2, 3… |
| Integers | Whole + negatives | … −2, −1, 0, 1, 2 … |
| Even | Divisible by 2 (ends 0,2,4,6,8) | 2, 4, 18 |
| Odd | Not divisible by 2 (ends 1,3,5,7,9) | 3, 7, 15 |
Division Algorithm:
Dividend = Divisor × Quotient + Remainder — remainder < divisor. Example: 13
÷4 → 13 = 4×3 +1 (quotient 3, remainder 1).
xy = 10x + y,
reverse yx = 10y + x.
(10x+y)−(10y+x)=9(x−y) → always divisible by 9. So difference
between a number and its reverse is a multiple of 9 — useful for "find the number" puzzles.
| Divisor | Rule |
|---|---|
| 2 | Last digit even (0,2,4,6,8) |
| 3 | Sum of digits divisible by 3 |
| 4 | Last two digits divisible by 4 |
| 5 | Last digit 0 or 5 |
| 6 | Divisible by both 2 and 3 |
| 7 | Double last digit, subtract from remaining truncated number; repeat until small — if result is 0 or divisible by 7, original is divisible. Eg 203: 20−2×3=14 → divisible by 7 |
| 8 | Last three digits divisible by 8 |
| 9 | Sum of digits divisible by 9 |
| 10 | Last digit is 0 |
| 11 | (Sum at odd places − sum at even places) is 0 or divisible by 11 |
| 12 | Divisible by both 3 and 4 |
| 13 | Add 4× last digit to remaining truncated number; repeat. Eg 3146: 314+4×6=338 → 33+4×8=65 → 65/13=5 → divisible |
| 14 | Divisible by both 2 and 7 |
| 15 | Divisible by both 3 and 5 |
| 16 | Last four digits divisible by 16 (since 16=2⁴). Shortcut: divisible by both 2 and 8 |
| 18 | Divisible by both 2 and 9 |
| 20 | Last digit 0 and second-last digit even (i.e., last two digits divisible by 20: 00,20,40,60,80) |
| 25 | Last two digits are 00, 25, 50, or 75 |
Pro tip for composites: To test 72, check co-prime factors 8 and 9 — if divisible by both, divisible by 72. Never use non-co-prime split (e.g., 6×12 for 72) — that gives false positives.
Any composite
N = p₁a × p₂b × p₃c × …
— its prime building blocks.
Example: 360 = 2³ × 3² × 5¹
(a+1)(b+1)(c+1)… — each prime can contribute 0 to a copies. For 360:
(3+1)(2+1)(1+1)=4×3×2=24 factors.
[(p₁a+1−1)/(p₁−1)] × [(p₂b+1−1)/(p₂−1)] × …
— geometric series per prime.
φ(N)=N×(1−1/p₁)×(1−1/p₂)×… — e.g., φ(360)=360×1/2×2/3×4/5=96. Needed for
Euler's theorem.
HCF and LCM: HCF = lowest powers of common primes; LCM = highest powers of
all primes. Golden rule: a×b = HCF×LCM (for two numbers only) — see Module 02.
n! = n × (n−1) × … × 1, 0! =1.
Count ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … — each multiple of p contributes one p,
multiples of p² contribute an extra.
Trailing zeros: a zero is 2×5; 2s are abundant, so count 5s via
Legendre. Zeros in n! = power of 5 in n!.
⌊100/5⌋=20 + ⌊100/25⌋=4 + ⌊100/125⌋=0 → 24 zeros. For
1000!: 200+40+8+1=249.
A ≡ R (mod M) means A leaves remainder R when divided by M. 13 o'clock ≡1 mod 12.
Last digit of a power cycles:
| Cycle | Bases (last digit) |
|---|---|
| Cycle 4 | 2, 3, 7, 8 (period 4) |
| Cycle 2 | 4, 9 (period 2) |
| Cycle 1 | 0, 1, 5, 6 (never changes) |
Method: divide exponent by cycle length, remainder gives effective exponent. Example:
7202 — 7 cycle 4 → 202 mod4=2 → last digit = 7²=49 → 9.
If remainder 0, use full cycle (power 4).
p and a not
divisible by p: ap−1 ≡1 mod p. So
ak mod p reduces via k mod (p−1). Example:
2100 mod101 → 2100≡1 mod101.
p:
(p−1)! ≡ −1 mod p (i.e., remainder p−1). Example: 10! mod11 → 10!
≡10.
n with gcd(a,n)=1:
aφ(n) ≡1 mod n — generalizes Fermat to composites. Example:
2φ(9)=2⁶=64≡1 mod9 (φ(9)=6).
Base 10 uses digits 0-9; other bases (binary 2, octal 8) are positional.
(123)₄ =1×4²+2×4¹+3×4⁰=16+8+3=27₁₀.
xy: Digits = ⌊y×log₁₀(x)⌋+1 (since log₁₀ gives magnitude). Example: digits in
2100 → ⌊100×0.3010⌋+1=31.
Sum of interior angles of n-gon = (n−2)×180° — e.g., hexagon 720°.
a²−b²=(a−b)(a+b)(a+b)²=a²+b²+2ab, (a−b)²=a²+b²−2abDividend = ?
Which is the only even prime?
Difference between a 2-digit number (10x+y) and its reverse (10y+x) is?
Divisible by 72 → check divisibility by?
Number of factors of 360=2³×3²×5¹?
Trailing zeros in 100! ?
Unit digit of 7²⁰² ?
Fermat: a^(p−1) mod p (prime p) = ?
Natural ⇒ 1…, Whole ⇒ 0…, Integer ⇒ ±whole ; Even→2| , Odd→¬
Dividend ⇒ Divisor×Quotient+Remainder
Prime ⇒ >1, 2 factors (2 only even) ; Composite ⇒ >2 ; Co-prime ⇒ GCD=1 (eg 8,9)
Digits ⇒ 10x+y vs 10y+x → diff 9(x−y) divisible by 9
2:even ; 3:sum%3 ; 4:last2%4 ; 5:0/5 ; 6:2&3 ; 7: 2×last subtract from rest ; 8:last3%8 ; 9:sum%9 ; 10:last 0
11:odd−even %11 ; 12:3&4 ; 13: 4×last + rest ; 14:2&7 ; 15:3&5 ; 16:last4%16 ; 18:2&9 ; 20:last2∈{00,20…80} ; 25:00/25/50/75
Composite ⇒ check co-prime factors (72→8&9)
N=p₁ap₂b… ; factors ⇒ (a+1)(b+1)… ; φ(N)=N∏(1−1/p)
HCF×LCM=a×b (2 nums only)
n! ⇒ product 1..n, 0!=1 ; Legendre: ⌊n/p⌋+⌊n/p²⌋… ; zeros ⇒ count 5s
Cyclicity: 2,3,7,8→cycle4 ; 4,9→2 ; 0,1,5,6→1 ; exp mod cycle
Fermat: ap−1≡1 (prime) ; Wilson: (p−1)!≡−1 ; Euler: aφ(n)≡1 (composite)
Base N→decimal: Σ digit×basepos ; digits in xy: ⌊y log₁₀x⌋+1 ; (n−2)×180° ; a²−b²=(a−b)(a+b)
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