HCF and LCM: Methods, Properties and Fractions
HCF and LCM: Methods, Properties and Fractions
HCF (Highest Common Factor) and LCM (Lowest Common Multiple) together form the most stable question source in NTPC Maths: the product rule, ratio-based LCM, LCM of fractions and decimals and HCF of algebraic expressions have all appeared in recent cycles, most recently in the March 2026 Graduate CBT-1 (the product rule and a pairwise-HCF minimisation).
1. Definitions and Methods
| Term | Meaning | Prime factorisation method |
|---|---|---|
| HCF (GCD) | Largest number dividing all given numbers | Common primes with the LOWEST powers |
| LCM | Smallest number divisible by all given numbers | All primes with the HIGHEST powers |
72 = 2³ × 3², 120 = 2³ × 3 × 5 → HCF = 2³ × 3 = 24, LCM = 2³ × 3² × 5 = 360
Division (Euclid) method for HCF: divide the larger by the smaller, then the previous divisor by the remainder, until the remainder is 0. The last divisor is the HCF. HCF(1651, 2032): 2032 = 1 × 1651 + 381; 1651 = 4 × 381 + 127; 381 = 3 × 127 + 0, so HCF = 127. Use it when the numbers are large and prime factors are not obvious.
Short-cut for HCF: the HCF of several numbers also divides every difference between them. HCF(403, 434, 465) must divide 31, and 31 divides all three, so HCF = 31.
2. Property Box
| Property | Statement |
|---|---|
| Product rule | HCF × LCM = product of the two numbers (ONLY for two numbers) |
| HCF divides LCM | LCM is always a multiple of HCF |
| Ratio rule | If numbers are in ratio a : b (lowest form) and HCF is h, numbers = ha and hb, LCM = hab |
| Co-primes | HCF = 1 and LCM = product |
| Consecutive numbers | Always co-prime; LCM of n and n + 1 = n(n + 1) |
| LCM of fractions | LCM of numerators / HCF of denominators |
| HCF of fractions | HCF of numerators / LCM of denominators |
| Decimals | Equalise decimal places, find HCF or LCM as whole numbers, put the places back |
| Powers of the same base | HCF(aᵐ − 1, aⁿ − 1) = a^HCF(m, n) − 1 |
| Algebraic expressions | Factorise; HCF = common factors with lowest powers, LCM = all factors with highest powers |
| Pairwise HCFs given | Each number is at least the LCM of the two HCFs it is part of; choose those, then verify every pairwise HCF |
✗ HCF × LCM = product of three numbers | ✓ The product rule holds only for two numbers
✗ LCM of fractions = LCM of numerators / LCM of denominators | ✓ It is LCM of numerators divided by HCF of denominators (fractions must be in lowest form)
Pairwise HCF check: the verification step matters. HCF(P, Q) = 6, HCF(Q, R) = 10, HCF(R, P) = 15 is impossible, because P must be a multiple of LCM(6, 15) = 30 and Q of LCM(6, 10) = 30, which makes HCF(P, Q) at least 30.
Counting co-prime pairs: if HCF = h and the sum is S, write the numbers as ha and hb with a + b = S/h and HCF(a, b) = 1, then count such pairs. Same idea with a product: ab = P/h².
हिंदी नोट: दो संख्याओं का म.स. × ल.स. = दोनों संख्याओं का गुणनफल। यह नियम केवल दो संख्याओं पर लागू होता है। भिन्नों का ल.स. = अंशों का ल.स. ÷ हरों का म.स.।
Exam Pointer: Verified NTPC patterns: the product rule (Graduate CBT-1, March 2026), LCM or HCF from a ratio (March 2021, June 2025), LCM of fractions and decimals (March 2021, August 2025 UG CBT-1), pairwise HCF minimisation (Graduate CBT-1, March 2026), and HCF of products or algebraic terms and LCM of roots (2020-21 cycle). The wrong-formula answer is the usual distractor, so a one-line check (HCF must divide LCM) eliminates traps.
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Product rule: find the other number
[PYQ: NTPC Graduate CBT-1 18-Mar-2026 Shift-3 | NTPC 2025-26 cycle]
EXAM LEVEL
Q. The HCF and LCM of two numbers are 12 and 2,016. If one number is 288, find the other.
Other number = (HCF × LCM)/given number = (12 × 2,016)/288 = 24,192/288 = 84. Check: HCF(84, 288) = 12, consistent.
Answer: 84
EXAMATLAS LEVEL
Q. The HCF of two numbers is 23 and their LCM is 1,449. If both numbers are three-digit numbers, find their sum.
Write the numbers as 23a and 23b with a, b co-prime and ab = 1,449/23 = 63. Co-prime pairs with product 63: (1, 63) and (7, 9). (1, 63) gives 23, which is two-digit, so the numbers are 23 × 7 = 161 and 23 × 9 = 207. Sum = 368. Picking (3, 21) fails because they are not co-prime (HCF would become 69).
Answer: 368
Pattern 2: LCM or HCF from a ratio
[PYQ: NTPC Graduate CBT-1 17-Jun-2025 Shift-1 | NTPC CBT-1 8-Mar-2021 Shift-2]
EXAM LEVEL
Q. Two numbers are in the ratio 3 : 5 and their HCF is 8. Find their LCM.
Numbers are 24 and 40. LCM = 8 × 3 × 5 = 120.
Answer: 120
EXAMATLAS LEVEL
Q. Two numbers are in the ratio 4 : 7 and their LCM is 1,008. Find their sum.
LCM = h × 4 × 7 = 28h = 1,008, so h = 36. Numbers 144 and 252, sum 396. Shortcut: sum = h × (4 + 7) = 36 × 11 = 396.
Answer: 396
Pattern 3: LCM and HCF of fractions and decimals
[PYQ: NTPC UG CBT-1 20-Aug-2025 Shift-2 | NTPC CBT-1 13-Mar-2021 Shift-1]
EXAM LEVEL
Q. Find the LCM of 2/3, 4/9 and 8/15.
LCM of numerators (2, 4, 8) = 8. HCF of denominators (3, 9, 15) = 3. LCM = 8/3.
Answer: 8/3
EXAMATLAS LEVEL
Q. Find the product of the HCF and the LCM of 1.08, 0.36 and 0.9.
Equalise to two decimal places: 1.08, 0.36, 0.90, which behave like 108, 36, 90. HCF(108, 36, 90) = 18, so HCF = 0.18. LCM(108, 36, 90) = 540, so LCM = 5.40. Product = 0.18 × 5.4 = 0.972. It is NOT the product of the three numbers, because the product rule needs exactly two numbers.
Answer: 0.972
Pattern 4: HCF and LCM of products and algebraic expressions
[PYQ: NTPC 2020-21 cycle]
EXAM LEVEL
Q. Find the HCF of 2³ × 3² × 5, 2² × 3³ × 7 and 2⁴ × 3 × 5².
Common primes are 2 and 3 only. Lowest powers: 2² and 3¹. HCF = 4 × 3 = 12.
Answer: 12
EXAMATLAS LEVEL
Q. Find the LCM of x² − 4, x² + x − 6 and x² − 5x + 6.
Factorise: x² − 4 = (x − 2)(x + 2), x² + x − 6 = (x + 3)(x − 2), x² − 5x + 6 = (x − 2)(x − 3). LCM takes every factor once at its highest power: (x − 2)(x + 2)(x + 3)(x − 3), which can also be written (x² − 4)(x² − 9). Their HCF is (x − 2). Writing (x − 2)³ in the LCM is the trap: a common factor appears only once, at its highest power in any single expression.
Answer: (x − 2)(x + 2)(x + 3)(x − 3)
Pattern 5: LCM of roots and HCF of powers
[PYQ: NTPC 2020-21 cycle]
EXAM LEVEL
Q. Find the LCM of √196, ∛125, ∜81 and √36.
The values are 14, 5, 3 and 6. LCM(14, 5, 3, 6) = 2 × 3 × 5 × 7 = 210.
Answer: 210
EXAMATLAS LEVEL
Q. Find the HCF of 2¹⁰⁰ − 1 and 2¹²⁰ − 1.
Use HCF(aᵐ − 1, aⁿ − 1) = a^HCF(m, n) − 1. HCF(100, 120) = 20, so the answer is 2²⁰ − 1 = 10,48,575.
Answer: 2²⁰ − 1
Pattern 6: Number of pairs with given HCF and sum or product
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. The sum of two numbers is 384 and their HCF is 48. How many such pairs exist?
Numbers 48a and 48b with a + b = 8 and a, b co-prime. Pairs: (1, 7) and (3, 5); (2, 6) and (4, 4) fail the co-prime test. So 2 pairs.
Answer: 2
EXAMATLAS LEVEL
Q. The product of two numbers is 2,028 and their HCF is 13. How many such pairs exist, and what are they?
Numbers 13a and 13b with ab = 2,028/169 = 12 and a, b co-prime. Pairs: (1, 12) and (3, 4); (2, 6) fails. Numbers: (13, 156) and (39, 52).
Answer: 2 pairs: (13, 156) and (39, 52)
Pattern 7: Pairwise HCFs given: smallest possible numbers
[PYQ: NTPC Graduate CBT-1 27-Mar-2026 Shift-3]
EXAM LEVEL
Q. For positive integers a, b and c, HCF(a, b) = 8, HCF(b, c) = 12 and HCF(c, a) = 4. Find the least possible value of a + b + c.
a shares 8 with b and 4 with c, so a is a multiple of LCM(8, 4) = 8. b shares 8 and 12, so it is a multiple of LCM(8, 12) = 24. c shares 12 and 4, so it is a multiple of 12. Try the smallest choices a = 8, b = 24, c = 12 and verify: HCF(8, 24) = 8, HCF(24, 12) = 12, HCF(12, 8) = 4, all correct. Least sum = 44.
Answer: 44
EXAMATLAS LEVEL
Q. HCF(P, Q) = 12, HCF(Q, R) = 18 and HCF(R, P) = 6. If P + Q + R is as small as possible, find P × Q × R.
P must be a multiple of LCM(12, 6) = 12, Q of LCM(12, 18) = 36 and R of LCM(18, 6) = 18. Smallest choice P = 12, Q = 36, R = 18. Verify before multiplying: HCF(12, 36) = 12, HCF(36, 18) = 18, HCF(18, 12) = 6, all match. Product = 12 × 36 × 18 = 7,776. Taking Q = 12 × 18 = 216 (product instead of LCM) still satisfies the HCFs but is not the smallest, which is the trap option.
Answer: 7,776
60-Second Revision
- HCF = lowest common powers; LCM = highest powers of all primes.
- HCF × LCM = product, for two numbers only; HCF always divides LCM.
- Ratio a : b with HCF h: numbers ha, hb; LCM = hab; sum = h(a + b).
- Fractions: LCM = LCM(num)/HCF(den); HCF = HCF(num)/LCM(den).
- Decimals: equalise places first; HCF(2ᵐ − 1, 2ⁿ − 1) = 2^HCF(m,n) − 1.
- Pairs with given HCF: reduce to co-prime a, b and count; pairwise HCFs: each number = LCM of its two HCFs, then verify.