Number System, HCF and LCM
Why This Chapter Matters
Number system is the foundation of SSC CGL Tier 1 and Tier 2 quantitative aptitude — 3-5 questions every paper. HCF, LCM, divisibility rules, and number properties are tested directly and as part of word problems.
Analogy — Think of HCF and LCM like finding the largest shared "building block" versus the smallest shared "meeting point" between two repeating patterns. HCF asks: what's the biggest identical piece both numbers can be cut into evenly? LCM asks: at what point do two repeating cycles (like two traffic lights blinking at different intervals) line up again at the same time? That's exactly why "bells ringing together" problems use LCM (finding the next shared meeting point) while "largest tile that evenly fits a floor" problems use HCF (finding the largest shared piece).
Core Concepts
1. Types of Numbers
Natural numbers: 1, 2, 3, ... (positive integers)
Whole numbers: 0, 1, 2, 3, ... (natural + zero)
Integers: ...-3, -2, -1, 0, 1, 2, 3...
Rational: p/q where q≠0 (includes decimals, fractions)
Irrational: cannot be expressed as p/q (√2, √3, π)
Real numbers: rational + irrational
Prime numbers: divisible only by 1 and itself (2 is the only even prime)
Composite numbers: more than 2 factors
2. Divisibility Rules
÷2: last digit even | ÷3: sum of digits divisible by 3
÷4: last 2 digits divisible by 4 | ÷5: ends in 0 or 5
÷6: divisible by both 2 and 3 | ÷7: complex rule (rarely tested directly)
÷8: last 3 digits divisible by 8 | ÷9: sum of digits divisible by 9
÷10: ends in 0 | ÷11: alternating digit sum difference divisible by 11
÷25: last 2 digits divisible by 25
3. HCF (Highest Common Factor)
HCF = largest number dividing all given numbers exactly.
Methods:
Prime factorisation: factorise each number → HCF = product of common prime factors with lowest powers.
Example: HCF(36, 48) → 36=2²×3², 48=2⁴×3 → HCF=2²×3=12
Division method (Euclid): divide larger by smaller → remainder becomes divisor → repeat.
HCF(48,36): 48=1×36+12 → 36=3×12+0 → HCF=12
4. LCM (Lowest Common Multiple)
LCM = smallest number divisible by all given numbers.
Formula: LCM × HCF = Product of two numbers (for exactly TWO numbers only)
LCM(a,b) = a×b / HCF(a,b)
Prime factorisation: LCM = product of all prime factors with highest powers.
LCM(36,48): 36=2²×3², 48=2⁴×3 → LCM=2⁴×3²=144
5. Important Properties
If HCF of a,b is H, then a=Hx, b=Hy where HCF(x,y)=1.
HCF always divides LCM. LCM is always a multiple of HCF.
HCF(a,b,c) divides each of a, b, c.
6. Word Problems Pattern
"Find largest number that divides x, y, z leaving remainder r each time"
→ HCF of (x-r), (y-r), (z-r)
"Find largest number dividing x leaving remainder p, y leaving remainder q"
→ HCF of (x-p) and (y-q)
"Find smallest number divisible by x, y, z" → LCM(x,y,z)
"Find smallest number that when divided by x, y, z leaves remainder r"
→ LCM(x,y,z) + r
"Bells ring at intervals of x, y, z minutes. When do they ring together?"
→ LCM(x,y,z) minutes after they ring together
Solved Examples
Q1: Find HCF and LCM of 12, 18, 24.
12=2²×3, 18=2×3², 24=2³×3
HCF=2×3=6 | LCM=2³×3²=72
Q2: The largest number that divides 245 and 1029 leaving remainder 5 each:
Numbers: 245-5=240, 1029-5=1024. HCF(240,1024).
240=2⁴×3×5, 1024=2¹⁰. HCF=2⁴=16.
Q3: Three bells ring at intervals of 9, 12, 15 minutes. If they ring together at 8 AM, when next?
LCM(9,12,15)=180 minutes=3 hours. Next: 11 AM.
PYQs (SSC CGL)
SSC CGL 2023: HCF of two numbers is 11, LCM is 7700. One number is 275. Find the other.
Other = HCF×LCM/First = 11×7700/275 = 308.
SSC CGL 2022: Find least number when divided by 5, 6, 7, 8 leaves remainder 3 in each case.
LCM(5,6,7,8)=840. Required=840+3=843.
SSC CHSL 2022: Which of the following is divisible by 11?
(A) 246521 (B) 415624 (C) 135792 (D) 358946
Alternating sum: (A) 2-2+5-6+4-2=1 ✗ (B) 4-2+6-5+1-4=0 ✓. Answer: B.
MCQ Practice
Q1. LCM of two numbers is 2310, HCF is 30. One number is 210. Other is?
(A) 230 (B) 330 ✓ (C) 120 (D) 310
[2310×30/210 = 330]
Q2. Greatest number dividing 29, 37, 53 leaving same remainder:
Differences: 37-29=8, 53-37=16, 53-29=24. HCF(8,16,24)=8. Answer: 8.
Q3 (Hard). A number when divided by 5 leaves 3, by 7 leaves 4. Find smallest such number.
Want: N≡3(mod5) and N≡4(mod7). Try N=5k+3: k=0→3, k=1→8, k=2→13, k=3→18, k=4→23, k=5→28, k=6→33 ✓ (33÷7=4r5 ✗), k=7→38 ✓ (38÷7=5r3 ✗).
Systematic: LCM(5,7)=35. Check 38: 38÷5=7r3✓, 38÷7=5r3✗. 53: 53÷5=10r3✓, 53÷7=7r4✓. Answer: 53.
Revision Notes
Try It (2 Minutes)
Two traffic lights blink every 12 seconds and every 18 seconds respectively, both starting together right now. Without looking up the formula, reason through when they'll next blink at exactly the same moment: list out the first few times each light blinks (12, 24, 36, 48... and 18, 36, 54...) and find the first number common to both lists. Check that this matches LCM(12,18) = 36 — and notice that you just derived LCM's actual meaning (the first shared point between two repeating cycles) rather than just applying a memorized formula.