Number System

Understand the family tree of numbers to avoid theoretical traps.

• Real Numbers: Everything on the number line.
• Rational (Q): Can be p/q (e.g., 2/3, 5). Terminating or Repeating decimals.
• Irrational: Non-terminating, Non-repeating (e.g., $\sqrt{2}, \pi$).
• Integers (Z): ...-2, -1, 0, 1, 2...
• Prime Numbers: Only two factors (1 and itself). 2 is the only even prime. 1 is NEITHER prime NOR composite.

To find the last digit of $a^n$, focus only on the unit digit of base and dividing power by 4.

• 0, 1, 5, 6: Same unit digit always. ($5^n \to 5$, $6^n \to 6$)
• 4, 9: Cycle of 2. ($4^{odd} \to 4, 4^{even} \to 6$; $9^{odd} \to 9, 9^{even} \to 1$)
• 2, 3, 7, 8: Cycle of 4. Divide power by 4, take remainder ($R$). If $R=0$, use power 4.

Check divisibility without dividing.

• 3 & 9: Sum of digits must be divisible by 3 or 9.
• 4: Last 2 digits divisible by 4.
• 8: Last 3 digits divisible by 8.
• 11: (Sum of Odd places) - (Sum of Even places) = 0 or multiple of 11.
• 7, 11, 13: Subtract last 3 digits from remaining part. Result divisible by 7/11/13?

Solve $a^n / d$ instantly.

• Negative Remainder: If remainder is large, take negative. (e.g., $17/18 o Rem -1$).
• Fermat's Little Theorem: $a^{p-1}/p o Rem 1$ (if p is prime).
• Wilson's Theorem: $(p-1)!/p o Rem (p-1)$.

Decompose numbers into prime factors ($N = a^p b^q c^r$).

• Total Factors: $(p+1)(q+1)(r+1)$.
• HCF: Lowest power of common primes.
• LCM: Highest power of all primes.
• Product Rule: $HCF imes LCM = N_1 imes N_2$.