1. Divisibility Rules

2. HCF & LCM

For two numbers, $\text{HCF} \times \text{LCM} = a \times b$, so $\text{LCM} = \dfrac{a\times b}{\text{HCF}}$ and $\text{HCF} = \dfrac{a\times b}{\text{LCM}}$.

Fractions: $\text{HCF} = \dfrac{\text{HCF of numerators}}{\text{LCM of denominators}}$ and $\text{LCM} = \dfrac{\text{LCM of numerators}}{\text{HCF of denominators}}$.

3. Prime Factorisation & Factors

If $N = p^{a}\,q^{b}\,r^{c}$ with $p, q, r$ prime:

• Total factors: $d(N) = (a+1)(b+1)(c+1)$
• Sum of factors: $\sigma(N) = \dfrac{p^{a+1}-1}{p-1}\cdot\dfrac{q^{b+1}-1}{q-1}\cdot\dfrac{r^{c+1}-1}{r-1}$
• Product of factors: $N^{\,d(N)/2}$
• Distinct prime factors: count the distinct primes (e.g. $2^{3}\cdot 3^{2}\cdot 5 \Rightarrow 3$)
• Odd factors: ignore the power of 2, then $(a_{\text{odd}}+1)(b_{\text{odd}}+1)\cdots$
• Even factors: total factors − odd factors

4. Factorials

$N! = N(N-1)(N-2)\cdots 2\cdot 1$, with $0! = 1! = 1$.

• Trailing zeros in $N!$: $Z = \left\lfloor\frac{N}{5}\right\rfloor + \left\lfloor\frac{N}{25}\right\rfloor + \left\lfloor\frac{N}{125}\right\rfloor + \cdots$
• Highest power of prime $p$ in $N!$ (Legendre): $\left\lfloor\frac{N}{p}\right\rfloor + \left\lfloor\frac{N}{p^{2}}\right\rfloor + \left\lfloor\frac{N}{p^{3}}\right\rfloor + \cdots$

5. Remainder & Divisibility Theorems

• $a^{n} - b^{n}$ is always divisible by $a - b$.
• $a^{n} + b^{n}$ is divisible by $a + b$ only when $n$ is odd.
• $a^{n} - b^{n}$ is divisible by $a + b$ only when $n$ is even.
• Fermat: if $p$ is prime and $\gcd(a,p)=1$, then $a^{p-1} \equiv 1 \pmod{p}$.
• Wilson: if $p$ is prime, $(p-1)! \equiv -1 \pmod{p}$.

6. Unit Digit (Cyclicity)

Only unit digits matter in a product — e.g. $37\times 48\times 29 \Rightarrow 7\times 8\times 9 = 504 \Rightarrow$ unit digit 4.

Use $n \bmod 2$ for length-2 digits and $n \bmod 4$ for length-4 (if the remainder is 0, use the last value in the cycle).

7. Number of Digits

Digits in $N$: $\lfloor \log_{10} N \rfloor + 1$. Digits in $N^{k}$: $\lfloor k\log_{10} N \rfloor + 1$.

8. Base Conversion

• Decimal → base $b$: repeated division by $b$ (read remainders bottom-up).
• Base $b$ → decimal: $(a_n a_{n-1}\dots a_0)_b = a_n b^{n} + a_{n-1} b^{\,n-1} + \cdots + a_0$.

9. Handy Properties

• $\gcd(n, n+1) = 1$ · $\gcd(2n, 2n+2) = 2$
• $n(n+1)$ is always even.
• $n(n+1)(n+2)$ is always divisible by 6.
• $n(n+1)(n+2)(n+3)$ is always divisible by 24.