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HCF and LCM for SSC CHSL

Bells that ring together, the largest tile that fits a floor exactly, the smallest number that leaves the same remainder: HCF and LCM questions in SSC CHSL come in a handful of familiar shapes. How to find HCF and LCM quickly and why the methods work, the product rule, fractions and decimals, how to tell which one a word problem wants, six worked questions and a practice set.

25 Sept 2026 7 min read

In this guide
  1. What HCF and LCM actually are
  2. Method 1: prime factorisation
  3. Method 2: the division (Euclid) method for HCF
  4. The product rule
  5. Fractions and decimals
  6. Which one does the question want?
  7. Six worked questions
  8. Common mistakes
  9. Practice
  10. What to do next

HCF (highest common factor) and LCM (lowest common multiple) questions are among the most predictable in SSC maths. The arithmetic is easy. What trips candidates up is choosing the wrong one: finding the LCM when the question wanted the HCF, or forgetting to adjust for a remainder. Once you can read a word problem and say "that's HCF" or "that's LCM" in two seconds, most of these take under a minute.

What HCF and LCM actually are

  • The HCF of two numbers is the largest number that divides both exactly. It can never be larger than the smaller number.
  • The LCM is the smallest number that both divide exactly. It can never be smaller than the larger number.

That size check alone eliminates wrong options surprisingly often.

Method 1: prime factorisation

Write each number as a product of primes.

36 = 2² × 3² and 48 = 2⁴ × 3.

  • HCF = the common primes, each at its lowest power = 2² × 3 = 12.
  • LCM = every prime that appears, each at its highest power = 2⁴ × 3² = 144.

Why: a common factor can't contain more 2s than the number with fewest 2s, so the lowest power is the most it can take. A common multiple must contain at least as many 2s as the number with most 2s, so the highest power is the least it needs.

Method 2: the division (Euclid) method for HCF

For larger numbers, repeatedly divide the larger by the smaller and replace the larger with the remainder, until the remainder is 0. The last divisor is the HCF.

HCF(624, 432): 624 = 1 × 432 + 192; 432 = 2 × 192 + 48; 192 = 4 × 48 + 0. HCF = 48.

Why: any number that divides both a and b also divides a − b, and so divides the remainder. The common divisors never change at any step, while the numbers get smaller.

The product rule

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

Why: for each prime, the HCF takes the lower power and the LCM takes the higher power. Between them they use both powers exactly once, just like the product.

Two useful consequences:

  • The HCF always divides the LCM. If an option gives HCF 12 and LCM 400, it is impossible, because 400 ÷ 12 is not a whole number.
  • If two numbers are in the ratio a : b (in lowest terms) and their HCF is h, the numbers are ah and bh, and their LCM is abh.

Fractions and decimals

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

(Reduce each fraction to lowest terms first.) Example: HCF of 2/3 and 4/9 = HCF(2, 4) ÷ LCM(3, 9) = 2/9. Check: 2/3 ÷ 2/9 = 3 and 4/9 ÷ 2/9 = 2, both whole numbers.

Decimals: make the number of decimal places equal, drop the point, work with whole numbers, then put the point back. HCF(0.6, 0.72) → HCF(60, 72) in hundredths = 12 → 0.12.

Which one does the question want?

The question asks for…UseTypical wording
The largest size, length or number that fits or divides exactlyHCF"largest tile", "longest tape", "greatest number that divides"
The first time things coincide againLCM"ring together again", "meet at the starting point"
The smallest number divisible by several numbersLCM"least number divisible by"
The smallest number leaving remainder r each timeLCM + r"leaves remainder 3 in each case"
The largest number leaving given remaindersHCF of (number − remainder)"leaves remainders 5 and 8 respectively"
The largest number leaving the same (unknown) remainderHCF of the differences"leaves the same remainder in each case"

Six worked questions

Q1. Three bells ring at intervals of 6, 8 and 12 minutes. They ring together at 9:00 a.m. When will they next ring together?
LCM(6, 8, 12): 6 = 2 × 3, 8 = 2³, 12 = 2² × 3, so LCM = 2³ × 3 = 24 minutes. They next ring together at 9:24 a.m.

Q2. A room is 6 m 24 cm long and 4 m 32 cm wide. Find the largest square tile that paves it exactly, and the number of such tiles.
In centimetres, 624 and 432. HCF = 48 cm (worked above). Tiles: (624 ÷ 48) × (432 ÷ 48) = 13 × 9 = 117 tiles.

Q3. The HCF of two numbers is 8 and their LCM is 240. One number is 48. Find the other.
8 × 240 = 48 × x, so x = 1,920 ÷ 48 = 40. Check: HCF(48, 40) = 8 and LCM = 240.

Q4. Find the largest number that divides 70 and 125, leaving remainders 5 and 8 respectively.
The number divides 70 − 5 = 65 and 125 − 8 = 117 exactly. HCF(65, 117): 117 = 1 × 65 + 52; 65 = 1 × 52 + 13; 52 = 4 × 13. So the answer is 13. Check: 70 = 5 × 13 + 5 and 125 = 9 × 13 + 8.

Q5. Find the largest number that divides 43, 91 and 183, leaving the same remainder in each case.
If all three leave remainder r, their differences are exact multiples of the divisor. The differences are 91 − 43 = 48, 183 − 91 = 92 and 183 − 43 = 140. HCF(48, 92) = 4 and HCF(4, 140) = 4, so the answer is 4. Check: all three leave remainder 3.

Q6. Find the greatest four-digit number divisible by 12, 15 and 18.
LCM: 12 = 2² × 3, 15 = 3 × 5, 18 = 2 × 3², so LCM = 2² × 3² × 5 = 180. 9,999 ÷ 180 = 55, remainder 99. So the answer is 9,999 − 99 = 9,900.

Common mistakes

MistakeFix
Using LCM for "largest tile""Largest … fits exactly" is always HCF
Forgetting to add the remainderSmallest number with remainder r = LCM + r
Subtracting the same remainder from each number when the remainders differSubtract each number's own remainder
Using the product rule for three numbersIt holds for two numbers only
Mixing units (m and cm)Convert everything to the smaller unit first

Practice

  1. Find the HCF and LCM of 18 and 30.
  2. The product of two numbers is 1,500 and their HCF is 10. Find their LCM.
  3. Traffic lights change every 30, 45 and 60 seconds. They change together at 10:00:00. When will they next change together?
  4. Find the largest number that divides 125, 175 and 225 exactly.
  5. Find the smallest number that leaves a remainder of 4 when divided by 6, 9 and 12.
  6. Find the LCM of 3/4 and 5/6.
  7. Two numbers are in the ratio 3 : 4 and their HCF is 5. Find their LCM.
  8. Two runners start together from the same point on a circular track. One completes a lap in 12 minutes and the other in 18 minutes. After how long are they next at the starting point together?

Answers:

  1. HCF 6, LCM 90. 18 = 2 × 3² and 30 = 2 × 3 × 5.
  2. 150. LCM = 1,500 ÷ 10.
  3. 10:03:00. LCM(30, 45, 60) = 180 seconds = 3 minutes.
  4. 25. 125 = 5³, 175 = 5² × 7, 225 = 3² × 5², so HCF = 5² = 25.
  5. 40. LCM(6, 9, 12) = 36, and 36 + 4 = 40.
  6. 15/2. LCM(3, 5) ÷ HCF(4, 6) = 15 ÷ 2. Check: 15/2 ÷ 3/4 = 10 and 15/2 ÷ 5/6 = 9.
  7. 60. The numbers are 15 and 20, and LCM = 3 × 4 × 5 = 60.
  8. 36 minutes. LCM(12, 18) = 36.

What to do next

  • Make the "which one?" table your own: write three wordings for each row.
  • Practise the division method on five pairs of three-digit numbers.
  • Solve 25 HCF–LCM questions from previous CHSL papers, timed at 45 seconds each.
  • Revise factors and divisibility in the number system guide, then move on to simplification.

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 Staff Selection Commission website .

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