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LCM and HCF for RRB NTPC: methods, patterns and practice

When will three bells ring together again? What is the largest number that divides three numbers leaving the same remainder? LCM and HCF questions follow a handful of fixed patterns. Each method with the reason it works, six worked questions and a practice set with answers.

25 Sept 2026 7 min read

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In this guide
  1. What the two terms mean
  2. Method 1: prime factorisation
  3. Method 2: repeated division for HCF
  4. The product rule, and its limit
  5. Which one does the question want?
  6. HCF and LCM of fractions
  7. Worked questions at NTPC level
  8. Common mistakes
  9. Practice set
  10. What to do next

LCM and HCF questions in RRB NTPC are short. Most are one of about six patterns: bells ringing together, the greatest length that measures several ropes, the smallest number that leaves a given remainder, and so on. Once you can tell which pattern you are looking at, the arithmetic takes seconds.

The topic also feeds others. Time and work, pipes and cisterns and fraction simplification all lean on LCM, so the time you spend here pays back later.

What the two terms mean

  • HCF (Highest Common Factor, also called GCD): the largest number that divides every given number exactly.
  • LCM (Lowest Common Multiple): the smallest number that every given number divides exactly.

A quick sense check follows straight from the definitions. The HCF can never be larger than the smallest number, and the LCM can never be smaller than the largest. The HCF always divides the LCM.

Method 1: prime factorisation

  1. Write each number as a product of prime powers.
  2. HCF = the primes common to all, each with its lowest power.
  3. LCM = every prime that appears, each with its highest power.

Why it works: a common factor can only use primes that every number has, and no more copies than the number with the fewest. A common multiple must contain every prime at least as many times as the number with the most.

For 36 = 2² × 3² and 84 = 2² × 3 × 7: HCF = 2² × 3 = 12, and LCM = 2² × 3² × 7 = 252.

Method 2: repeated division for HCF

Divide the larger number by the smaller, then divide the divisor by the remainder, and repeat until the remainder is 0. The last divisor is the HCF.

For 84 and 36: 84 = 2 × 36 + 12, then 36 = 3 × 12 + 0. HCF = 12.

Why it works: any number that divides both 84 and 36 must also divide 84 − 2 × 36 = 12. So the pair (84, 36) has exactly the same common factors as the pair (36, 12), and the numbers keep shrinking until the answer is obvious. This is faster than factorising when the numbers are large, such as 1,071 and 462.

The product rule, and its limit

For two numbers only: HCF × LCM = product of the two numbers.

Why it works: for each prime, the HCF takes the smaller power and the LCM takes the larger. Smaller plus larger equals the two powers added together, which is exactly what the product has.

With three numbers this breaks, because the HCF and LCM take only the extreme powers and skip the middle one. For 2, 4 and 8: HCF × LCM = 2 × 8 = 16, but the product is 64.

Which one does the question want?

The question saysUseReason
"together again", "at the same time", "smallest number divisible by"LCMYou need a number every interval fits into
"largest number that divides", "greatest length", "maximum size of tiles", "fewest equal pieces"HCFYou need a size that fits into every quantity
"leaves the same remainder r" (smallest number)LCM + rTake a multiple of all, then add r
"leaves the same remainder" (largest divisor, remainder unknown)HCF of the differencesThe remainder cancels when you subtract
"leaves remainders r₁, r₂" (largest divisor)HCF of (x − r₁) and (y − r₂)Removing each remainder makes the numbers exact multiples
Remainders each fall short of the divisor by the same kLCM − kThe number is k less than a common multiple

HCF and LCM of fractions

  • HCF of fractions = HCF of the numerators ÷ LCM of the denominators.
  • LCM of fractions = LCM of the numerators ÷ HCF of the denominators.

For 2/3, 4/9 and 8/15: HCF = 2/45 and LCM = 8/3. Why: to divide into every fraction, the answer needs a small top and a large bottom, and the reverse holds for a multiple. Reduce each fraction to lowest terms first.

Worked questions at NTPC level

Q1. Three bells toll at intervals of 12, 18 and 30 minutes. They toll together at 8:00 a.m. When will they next toll together?

12 = 2² × 3, 18 = 2 × 3², 30 = 2 × 3 × 5. LCM = 2² × 3² × 5 = 180 minutes = 3 hours.
They toll together again at 11:00 a.m.

Q2. Find the largest number that divides 43, 91 and 183, leaving the same remainder in each case.

Differences: 91 − 43 = 48, 183 − 91 = 92, 183 − 43 = 140.
HCF of 48, 92 and 140 = 4.
Check: 43, 91 and 183 each leave remainder 3 when divided by 4.

Q3. Find the smallest number which, when divided by 6, 9 and 15, leaves remainders 2, 5 and 11 respectively.

Look at the gaps: 6 − 2 = 4, 9 − 5 = 4, 15 − 11 = 4. Every remainder is 4 short of its divisor, so the number is 4 less than a common multiple.
LCM of 6, 9 and 15 = 90. Answer: 90 − 4 = 86.
Check: 86 = 6 × 14 + 2 = 9 × 9 + 5 = 15 × 5 + 11.

Q4. Three rods measure 4 m 95 cm, 9 m and 16 m 65 cm. Find the greatest length of a scale that can measure all three exactly.

Convert to one unit: 495 cm, 900 cm and 1,665 cm.
495 = 3² × 5 × 11; 900 = 2² × 3² × 5²; 1,665 = 3² × 5 × 37.
HCF = 3² × 5 = 45 cm.

Q5. The product of two numbers is 2,160 and their HCF is 12. How many such pairs of numbers are possible?

Write the numbers as 12a and 12b, where a and b share no factor. Then 144ab = 2,160, so ab = 15.
Co-prime pairs with product 15: (1, 15) and (3, 5). So 2 pairs: 12 and 180, or 36 and 60.

Q6. Two numbers are in the ratio 3 : 4 and their LCM is 180. Find their HCF.

Let the numbers be 3x and 4x, with x their HCF. Since 3 and 4 share no factor, LCM = 12x.
12x = 180, so x = 15. The numbers are 45 and 60. Check: HCF(45, 60) = 15, LCM = 180.

Common mistakes

  • Using LCM when the question says "largest". That wording almost always means HCF.
  • Applying HCF × LCM = product to three numbers. It holds for two only.
  • Forgetting the remainder. In "leaves remainder 4 in each case", the answer is LCM + 4, not the LCM.
  • Mixing units. Convert metres and centimetres, or hours and minutes, before you factorise.
  • Accepting impossible data. If a question gives HCF 12 and LCM 250, stop: 12 does not divide 250, so no such pair exists.

Practice set

  1. Find the LCM of 12, 18 and 30.
  2. Find the HCF of 48 and 180.
  3. The HCF of two numbers is 8 and their LCM is 96. One number is 24. Find the other.
  4. Three traffic lights change every 30, 45 and 60 seconds. They change together now. After how long will they next change together?
  5. Find the largest number that divides 70 and 125, leaving remainders 5 and 8 respectively.
  6. Find the largest four-digit number divisible by 12, 15 and 18.
  7. Find the smallest number which leaves remainder 3 when divided by 8, 12 or 16.
  8. Find the HCF and LCM of 2/3, 4/9 and 8/15.

Answers:

  1. 180. 12 = 2² × 3, 18 = 2 × 3², 30 = 2 × 3 × 5; LCM = 2² × 3² × 5.
  2. 12. 180 = 3 × 48 + 36; 48 = 1 × 36 + 12; 36 = 3 × 12.
  3. 32. Other number = 8 × 96 ÷ 24.
  4. 180 seconds (3 minutes). LCM of 30, 45 and 60.
  5. 13. HCF of 70 − 5 = 65 and 125 − 8 = 117. 65 = 5 × 13, 117 = 9 × 13.
  6. 9,900. LCM = 180. 9,999 ÷ 180 leaves remainder 99, so 9,999 − 99 = 9,900.
  7. 51. LCM of 8, 12 and 16 is 48; 48 + 3 = 51.
  8. HCF 2/45, LCM 8/3. HCF(2, 4, 8) ÷ LCM(3, 9, 15) = 2/45; LCM(2, 4, 8) ÷ HCF(3, 9, 15) = 8/3.

What to do next

  • Copy the "which one does the question want" table onto a card and read it before each practice session this week.
  • Solve 25 past NTPC LCM–HCF questions with a timer, aiming for 45 seconds each.
  • Revise prime factors in the number system guide if factorising feels slow.
  • Move on to time and work, where the LCM method does most of the work.

A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Railway Recruitment Boards website .

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