Sniper Mode: Numbers

1. Classification of Numbers

• Natural (N): 1, 2, 3... (Countable).
• Whole (W): 0 + N.
• Integers (Z): ...-2, -1, 0, 1, 2...
• Rational (Q): p/q form (q!=0). Ex: 2/3, 5 (5/1), 0 (0/1).
• Irrational: Cannot be p/q. Ex: √2, √3, π (Pi).
• Prime: Divisible by 1 and itself. (2, 3, 5, 7, 11...). 2 is only Even Prime.
• Composite: More than 2 factors. (4, 6...). 1 is neither Prime nor Composite.

2. Divisibility Rules (Sniper Checks)

• 2: Last digit even.
• 3: Sum of digits divisible by 3.
• 4: Last 2 digits divisible by 4.
• 5: Last digit 0 or 5.
• 6: Divisible by 2 AND 3.
• 8: Last 3 digits divisible by 8.
• 9: Sum of digits divisible by 9.
• 11: (Sum of Odd places) - (Sum of Even places) = 0 or 11k.
• 7/13: Take groups of 3 from right. Differences of sums. (Hardly used, rely on direct division).

3. Unit Digit Concept

• Cyclicity 1: 0, 1, 5, 6 (Same unit digit). (Ex: 5^123 -> 5).
• Cyclicity 2: 4 (4,6), 9 (9,1).
• Cyclicity 4: 2, 3, 7, 8. (Divide power by 4, take remainder).
  • Remainder 1 -> Power 1.
  • Remainder 0 -> Power 4.

4. Remainder Theorem

• Dividend = Divisor x Quotient + Remainder.
• Negative Remainder: 17/5 -> Remainder is +2 or -3. Use -ve to simplify calculation.

5. VBODMAS

• V: Vinculum (Bar).
• B: Brackets ().
• O: Of (x).
• D: Divide.
• M: Multiply.
• A: Add.
• S: Subtract.