Number System Divisibility LCM HCF Questions for UP POLICE CONSTABLE

Divisibility Rules — Quick Checks

Divisibility rules let you instantly check if a number divides evenly without actually dividing. In UP Police exams, you'll see questions like 'Which number is divisible by 6?' or 'Find the largest number divisible by 12 that is less than 500.' Knowing these rules saves calculation time and reduces errors. Think of them as shortcuts — like checking if a door is locked before trying to open it.

• Divisible by 2: Last digit is even (0, 2, 4, 6, 8)
• Divisible by 3: Sum of all digits is divisible by 3
• Divisible by 4: Last two digits form a number divisible by 4
• Divisible by 5: Last digit is 0 or 5
• Divisible by 6: Divisible by both 2 AND 3
• Divisible by 8: Last three digits form a number divisible by 8
• Divisible by 9: Sum of all digits is divisible by 9
• Divisible by 11: Alternating sum of digits is divisible by 11 (e.g., 1-2+3-4 for 1234)

HCF (Highest Common Factor) — Finding Common Divisors

HCF is the largest number that divides two or more numbers exactly. In UP Police exams, HCF questions appear as: 'What is the HCF of 48 and 64?' or 'Divide 120 and 180 sweets equally among children — maximum children possible?' The key insight: HCF solves 'maximum grouping' problems. Use prime factorization or Euclidean algorithm (repeated division) — both are fast.

• HCF is always less than or equal to the smallest number
• Prime factorization method: Find common prime factors, multiply them once
• Euclidean algorithm: Divide larger by smaller, then divide divisor by remainder, repeat until remainder is 0
• HCF of coprime numbers (no common factors) is always 1
• HCF is used in 'maximum equal distribution' problems

LCM (Least Common Multiple) — Finding Common Multiples

LCM is the smallest number that is a multiple of two or more numbers. In UP Police exams, LCM questions appear as: 'Bells ring every 6, 8, and 12 seconds — when do they ring together?' or 'Find the smallest number divisible by 4, 5, and 6.' The key insight: LCM solves 'minimum synchronization' problems. Use prime factorization (take highest power of each prime) or the formula LCM × HCF = a × b.

• LCM is always greater than or equal to the largest number
• Prime factorization method: Take highest power of each prime factor
• LCM of coprime numbers = their product
• LCM is used in 'minimum common occurrence' problems (bells, meetings, cycles)
• For two numbers: LCM(a, b) = (a × b) / HCF(a, b)

Word Problem Patterns — Recognizing When to Use HCF vs LCM

UP Police exams test your ability to identify whether a problem needs HCF or LCM. The language of the question is your clue. 'Maximum' or 'largest' or 'divide equally' = HCF. 'Minimum' or 'smallest' or 'together again' or 'at the same time' = LCM. Misidentifying this costs you the entire question.

• HCF keywords: maximum, largest, divide equally, distribute, common divisor, greatest
• LCM keywords: minimum, smallest, together, same time, again, common multiple, least
• HCF example: 'Maximum children to share 120 apples and 180 oranges equally' → HCF(120, 180) = 60 children
• LCM example: 'Two buses depart every 12 and 18 minutes. When do they depart together?' → LCM(12, 18) = 36 minutes
• Always read the question twice — once for understanding, once to identify HCF or LCM

• Confusing HCF and LCM: HCF is SMALLER (divides into numbers), LCM is LARGER (numbers divide into it). Use the analogy: HCF = common ancestor (one person), LCM = common descendant (many people).
• Forgetting to check divisibility rules before calculating: Many aspirants jump to long division when a quick divisibility check would save 20 seconds. Always check 2, 3, 5 first.
• In Euclidean algorithm, stopping too early: Continue dividing until remainder is exactly 0, not just 'small'. A common error is stopping at remainder 1 and claiming HCF = 1.
• Misidentifying prime factors: When finding LCM, forgetting to take the HIGHEST power of each prime. Example: LCM(12, 18) = 2 × 3 × 6 (wrong) vs 2^2 × 3^2 = 36 (correct).
• Not simplifying before calculating: If asked for HCF(100, 150), divide both by 10 first to get HCF(10, 15) = 5, then multiply back by 10 to get 50. Saves mental effort.

• HCF = 'H' for 'High-est' but it's the SMALLER number. LCM = 'L' for 'Least' but it's the LARGER number. Opposite naming = easy to mix up. Mnemonic: 'HCF divides IN, LCM divides OUT.'
• Divisibility by 11: Alternating sum. Example 1234: 1 - 2 + 3 - 4 = -2 (not divisible by 11). Think of it as a 'zigzag' pattern.
• LCM × HCF = a × b. This is your emergency formula if you forget the prime factorization method. Always works for two numbers.
• Maximum = HCF, Minimum = LCM. Read the question, spot the keyword, choose the method. This is 80% of word problem success.

• UP Police papers typically include 2-3 pure divisibility/HCF/LCM questions and 1-2 word problems. Total 5-8 minutes if you know the rules; 15+ minutes if you don't. Prioritize mastering divisibility rules and the LCM-HCF relationship.
• Recent papers show a trend toward 'find the smallest/largest number' questions combined with divisibility. Example: 'Smallest 4-digit number divisible by 6, 8, 12.' Answer: LCM(6, 8, 12) = 24, then find smallest 4-digit multiple of 24.
• Euclidean algorithm is faster than prime factorization for large numbers (e.g., HCF of 1071 and 462). Practice this method — it appears in 1-2 questions per paper.
• Watch for 'three numbers' questions (HCF/LCM of 12, 18, 24). Prime factorization is cleaner here than repeated Euclidean steps.
• Time management: Divisibility checks should take 10-15 seconds per question. If you're taking longer, you're calculating instead of checking. Use the rules, not long division.