CUET UG General Test Quantitative Reasoning Number System HCF LCM Divisibility Rules Surds

Must‑Know (15–20 detailed bullets)

• HCF of two or more numbers is the largest number that divides each of them exactly. Example: HCF of 12 and 18 is 6.
• LCM of two or more numbers is the smallest number that is exactly divisible by each of them. Example: LCM of 4 and 6 is 12.
• Product of two numbers = Product of their HCF and LCM. For 12 and 18: 12 × 18 = 216 = 6 × 36.
• If HCF of two numbers is 1, they are co-prime. Example: 8 and 15 are co-prime.
• For fractions: HCF = (HCF of numerators) / (LCM of denominators). Example: HCF of 2/3 and 4/5 = HCF(2,4)/LCM(3,5) = 2/15.
• LCM of fractions = (LCM of numerators) / (HCF of denominators). Example: LCM of 2/3 and 4/5 = LCM(2,4)/HCF(3,5) = 4/1 = 4.
• Divisibility rule for 2: last digit even. Example: 246 is divisible by 2.
• Divisibility rule for 3: sum of digits divisible by 3. Example: 123 → 1+2+3=6 → divisible by 3.
• Divisibility rule for 4: last two digits form a number divisible by 4. Example: 1324 → 24 ÷ 4 = 6 → divisible.
• Divisibility rule for 5: last digit 0 or 5. Example: 105 ends in 5 → divisible.
• Divisibility rule for 6: divisible by both 2 and 3. Example: 132 → even and 1+3+2=6 → divisible by 6.
• Divisibility rule for 8: last three digits divisible by 8. Example: 1232 → 232 ÷ 8 = 29 → divisible.
• Divisibility rule for 9: sum of digits divisible by 9. Example: 234 → 2+3+4=9 → divisible.
• Divisibility rule for 10: last digit 0. Example: 120 → divisible.
• Surds are irrational numbers that cannot be simplified to remove a root. Example: √2, √3 are surds; √4 = 2 is not.
• √a × √b = √(ab). Example: √3 × √5 = √15.
• √a / √b = √(a/b). Example: √12 / √3 = √4 = 2.
• Rationalizing factor of √a is √a; product = a. Example: √5 × √5 = 5.
• Rationalizing factor of (a + √b) is (a – √b). Example: (2 + √3)(2 – √3) = 4 – 3 = 1.
• If a number leaves remainder r when divided by d, then number = dq + r, where 0 ≤ r < d. Example: 23 = 5×4 + 3.

Difficulty Level

Intermediate — requires application of multiple rules and number manipulation, but no advanced theorems.

Common CUET Traps (3 bullets)

• Trap: Assuming LCM of fractions uses HCF of denominators instead of LCM.
  Avoid: Use LCM of numerators / HCF of denominators for LCM of fractions.

• Trap: Confusing HCF and LCM in word problems involving "minimum time" or "least number".
  Avoid: "Least" usually means LCM; "greatest" or "maximum size" usually means HCF.

• Trap: Treating √(a + b) as √a + √b.
  Avoid: √(a + b) ≠ √a + √b. Example: √(9 + 16) = √25 = 5 ≠ 3 + 4 = 7.

Practice MCQs (5 questions)

Q1. What is the HCF of 36 and 84?
A) 6
B) 12
C) 18
D) 24
Answer: B) 12
Explanation: 36 = 2² × 3², 84 = 2² × 3 × 7 → HCF = 2² × 3 = 12.
Why others fail: 6 is a common factor but not the highest.

Q2. Which of the following is divisible by 9?
A) 1234
B) 2345
C) 3456
D) 4567
Answer: C) 3456
Explanation: 3+4+5+6=18, divisible by 9 → 3456 is divisible by 9.
Why others fail: 4567 sums to 22, not divisible by 9 — distracts with higher digits.

Q3. The LCM of two numbers is 240 and their HCF is 12. If one number is 60, what is the other?
A) 48
B) 54
C) 60
D) 72
Answer: A) 48
Explanation: Product = HCF × LCM → 60 × x = 12 × 240 → x = (12×240)/60 = 48.
Why others fail: Students may divide 240 by 60 and get 4, then multiply by 12 incorrectly.

Q4. On simplifying (√5 + √3)(√5 – √3), the result is:
A) 2
B) 8
C) √2
D) 5 – 3√15
Answer: A) 2
Explanation: (√5 + √3)(√5 – √3) = (√5)² – (√3)² = 5 – 3 = 2.
Why others fail: Option D distracts with incorrect expansion using distributive property wrongly.

Q5. Three bells toll at intervals of 12, 15, and 18 minutes respectively. If they start together, after how many minutes will they toll together again?
A) 90
B) 120
C) 180
D) 240
Answer: C) 180
Explanation: LCM of 12, 15, 18 = LCM(2²×3, 3×5, 2×3²) = 2²×3²×5 = 180.
Why others fail: 90 is LCM of 15 and 18 but not divisible by 12.

Last‑Minute Revision (15–20 one‑liners)

• ⚠️ HCF × LCM = Product of two numbers — valid only for two numbers.
• ⚠️ Co-prime numbers have HCF = 1. Example: (8,9).
• ⚠️ LCM of co-prime numbers = their product. Example: LCM(7,11) = 77.
• ⚠️ Divisibility by 7: double the last digit, subtract from rest — if result divisible by 7, so is original number. Example: 161 → 16 – 2 = 14 → divisible.
• ⚠️ √ab = √a × √b — true only for non-negative a, b.
• ⚠️ √(a/b) = √a / √b — valid for a ≥ 0, b > 0.
• ⚠️ Rationalizing factor of √a + √b is √a – √b — difference of squares eliminates roots.
• ⚠️ 0 is divisible by every integer except 0 itself.
• ⚠️ 1 is not a prime number — it has only one factor.
• ⚠️ Prime numbers > 3 are of the form 6n ± 1 — verify from NCERT.
• ⚠️ HCF of fractions = HCF(numerators)/LCM(denominators).
• ⚠️ LCM of fractions = LCM(numerators)/HCF(denominators).
• ⚠️ Divisibility by 11: (sum of digits at odd places) – (sum at even places) divisible by 11. Example: 121 → (1+1) – 2 = 0 → divisible.
• ⚠️ √2 ≈ 1.414, √3 ≈ 1.732 — useful for approximation.
• ⚠️ Surds cannot be expressed as exact decimals — they are irrational.
• ⚠️ When dividing by d, remainder is always less than d.
• ⚠️ Successive division: remainder depends on order — not commutative.
• ⚠️ Use prime factorization to find HCF and LCM — most reliable method.
• ⚠️ Mnemonic: "HCF = Highest Common Factor → take lowest powers of common primes."
• ⚠️ Mnemonic: "LCM = Least Common Multiple → take highest powers of all primes."