Number System Divisibility LCM HCF Questions for BIHAR POLICE CONSTABLE

Divisibility Rules (Quick Checks)

Divisibility rules let you instantly check if a number divides another without long division. In Bihar Police exams, these appear as standalone 'which number divides X' questions and as filters in HCF/LCM problems. Learning these saves massive time during calculation.

• 2: Last digit is even (0, 2, 4, 6, 8)
• 3: Sum of digits divisible by 3 (e.g., 573: 5+7+3=15, divisible by 3)
• 4: Last two digits form a number divisible by 4 (e.g., 316: 16 is divisible by 4)
• 5: Last digit is 0 or 5
• 6: Divisible by both 2 AND 3 (e.g., 18 is even and 1+8=9 divisible by 3)
• 8: Last three digits form a number divisible by 8 (e.g., 5216: 216÷8=27)
• 9: Sum of digits divisible by 9 (e.g., 729: 7+2+9=18, divisible by 9)
• 11: Alternating sum of digits divisible by 11 (e.g., 1331: 1-3+3-1=0, divisible by 11)

HCF (Highest Common Factor) / GCD

HCF is the largest number that divides two or more numbers exactly. In Bihar Police exams, HCF appears in 'divide into equal groups' problems: e.g., 'How many maximum equal groups can 48 apples and 36 oranges be divided into?' Answer: HCF(48,36)=12. Two methods: Prime Factorization (slower, reliable) and Euclidean Algorithm (faster for large numbers).

• Prime Factorization: Find common prime factors, multiply lowest powers (e.g., 48=2^4×3, 36=2^2×3^2, HCF=2^2×3=12)
• Euclidean Algorithm: HCF(a,b) = HCF(b, a mod b) repeated until remainder=0 (faster for exam)
• HCF of coprime numbers (no common factors) is always 1
• HCF is always less than or equal to the smallest number
• Word problem clue: 'maximum equal', 'divide equally', 'largest group' → use HCF

LCM (Least Common Multiple)

LCM is the smallest number divisible by two or more numbers. In Bihar Police exams, LCM appears in 'meet again' or 'repeat cycle' problems: e.g., 'Two buses leave every 12 and 18 minutes. When do they leave together again?' Answer: LCM(12,18)=36 minutes. Prime Factorization method: multiply all prime factors with highest powers.

• Prime Factorization: List all prime factors, take highest power of each (e.g., 12=2^2×3, 18=2×3^2, LCM=2^2×3^2=36)
• LCM is always greater than or equal to the largest number
• LCM of coprime numbers is their product
• Word problem clue: 'together again', 'same time', 'next meeting', 'repeat' → use LCM
• For 3+ numbers: find LCM(a,b) first, then LCM(result, c)

Mixed HCF & LCM Word Problems

Bihar Police exams often combine HCF and LCM in one problem. The key is identifying which operation solves which part. Read carefully: 'divide equally' or 'maximum' → HCF; 'together again' or 'repeat' → LCM. Some problems ask for both.

• Two-step problems: Find HCF first to divide, then LCM to find repeat cycle
• Remainder problems: If 'remainder r when divided by a, b, c', find LCM(a,b,c) - r
• Ratio problems: HCF helps simplify ratios to lowest terms
• Exam trick: Read the question twice—sometimes 'maximum' hides in the middle

• Confusing HCF and LCM: Remember HCF is smaller (High Common Factor is high in dividing power, low in value); LCM is larger. Use the words: HCF = 'divide', LCM = 'multiple'.
• Forgetting to use divisibility rules: Aspirants waste time on long division when divisibility rules give instant answers. Memorize the 8 rules above.
• Wrong prime factorization: Mistakes in breaking numbers into primes cascade into wrong HCF/LCM. Double-check: 36 = 2^2 × 3^2, NOT 2 × 3^2.
• Misreading word problems: 'Divide 48 apples and 36 oranges into maximum equal groups' is HCF, not LCM. Aspirants panic and pick the wrong operation.
• Forgetting the HCF × LCM = a × b identity: This is a lifesaver when one value is missing. Many exams test this shortcut.

• HCF = 'Highest Common Factor' = divide down = smaller number. LCM = 'Least Common Multiple' = multiply up = larger number.
• HCF clues: 'max', 'equal', 'divide', 'group', 'largest'. LCM clues: 'together', 'again', 'next', 'repeat', 'cycle'.
• Divisibility 3 & 9: Sum of digits. Divisibility 11: Alternating sum. (3 and 9 are 'sum friends'; 11 is 'alternating friend'.)
• HCF × LCM = a × b: This is the golden identity. If you forget how to compute one, use this.

• Bihar Police Constable papers (2022-2024) show 2-3 direct divisibility/HCF/LCM questions in the 20-30 minute numerical section. Expect 1 pure divisibility rule check, 1 HCF word problem, 1 LCM or mixed problem.
• Time pressure is real: Divisibility rules are your speed weapon. A question like 'Is 5832 divisible by 9?' takes 5 seconds with the rule (5+8+3+2=18, yes) vs. 20 seconds with division.
• Word problems are tricky: Re-read once. Underline 'maximum', 'equal', 'together', 'again'. These keywords decide HCF vs. LCM instantly.
• Recent trend: Exams mix HCF and LCM in one problem, e.g., 'Find HCF, then use it to find LCM'. Practice combined problems.
• Calculator is not allowed: Master Euclidean Algorithm for HCF of large numbers (e.g., HCF(1071, 462)) and prime factorization for LCM. Both are faster than long division under exam stress.