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HCF and LCM Word Problems: Deciding Which One the Question Needs

Bells and lights (LCM) • Tiles and measures (HCF) • Same remainder • Different remainders • HCF × LCM = product

SSC Banking Railway 📅 Last updated: 2026-10-20

Almost every HCF or LCM word problem can be classified with a single question: will the answer be smaller or larger than the numbers in the question?

  • Smaller, so use HCF. You are cutting, dividing, packing or measuring several quantities into equal pieces, and you want the biggest possible piece.
  • Larger, so use LCM. You are waiting for repeating events to line up again, or looking for a number that several divisors all fit into.

The calculation is rarely the hard part. The hard part is reading the situation correctly. The sections below work through the six patterns that appear most often, each with a worked example.

Bells, traffic lights, runners on a circular track and buses leaving a depot all repeat at fixed intervals. They come together again after a time that is a multiple of every interval, and the first such time is the LCM.

Worked example 1: three lights

Three signal lights change every 6, 8 and 12 minutes and change together at noon. They next change together after LCM(6, 8, 12) = 24 minutes, at 12:24 p.m.

The same idea finds the smallest number with given divisors. If the question adds a condition such as "four-digit", find the LCM and take its first multiple that satisfies the condition.

Worked example 2: the smallest four-digit multiple

The smallest four-digit number divisible by 12, 15 and 18: the LCM is 180, and the first multiple of 180 above 999 is 6 × 180 = 1,080.

When you divide several lengths, volumes or groups into equal parts without any left over, the size of each part must divide every quantity. The largest such size is the HCF. Questions about the minimum number of pieces use the same HCF, followed by a division.

Worked example 3: square tiles

A floor measures 12.6 m by 8.4 m. The largest square tile that fits exactly has side HCF(1,260 cm, 840 cm) = 420 cm. Converting to centimetres first avoids decimals.

The number of such tiles is (1,260 ÷ 420) × (840 ÷ 420) = 3 × 2 = 6. A larger tile would not fit exactly, and a smaller one would need more tiles.

Remainder questions are where most mistakes happen, because each variety needs a slightly different adjustment. There are three.

WordingMethod
Least number leaving the same remainder rLCM + r
Least number where each remainder is the divisor minus kLCM − k
Greatest number dividing several numbers with the same remainderHCF of the differences
Greatest number dividing numbers with given remaindersHCF of (each number − its remainder)

Worked example 4: LCM plus the remainder

The least number that leaves remainder 5 when divided by 12, 15 and 20 is LCM(12, 15, 20) + 5 = 60 + 5 = 65.

Worked example 5: each remainder two short

Find the least number that leaves remainders 3, 4 and 6 when divided by 5, 6 and 8. Each remainder is 2 less than its divisor, so the number is 2 short of a common multiple: LCM(5, 6, 8) − 2 = 120 − 2 = 118.

Worked example 6: equal remainders, greatest divisor

The greatest number dividing 43, 91 and 183 with the same remainder divides every difference. The differences are 48, 92 and 140, and their HCF is 4.

Worked example 7: given remainders

The greatest number dividing 1,657 and 2,037 with remainders 6 and 5 must divide 1,651 and 2,032 exactly. Their HCF is 127.

For any two numbers, HCF × LCM = the product of the numbers. Questions give three of the four values and ask for the fourth. Remember that the rule applies only to two numbers.

Worked example 8: the missing number

Two numbers have HCF 12 and LCM 180, and one of them is 36. The other is 12 × 180 ÷ 36 = 60.

A quick sanity check: the HCF must divide both numbers and the LCM must be a multiple of both. Here 12 divides 36 and 60, and 180 is a multiple of each, so the answer is consistent.

  • ✘ Using HCF for "when will they meet again". Meeting again means waiting, which is LCM.
  • ✘ Forgetting to convert metres and centimetres to one unit before taking the HCF.
  • ✘ Applying HCF × LCM = product to three numbers.
  • ✘ Adding the remainder in "greatest number" questions. There you subtract it from each number.

Fast divisibility tests make finding HCFs easier; see divisibility rules. Coinciding events also appear in the LCM method for time and work, and the formula list is in the Quant cheat code.

Q1. Three bells ring at intervals of 9, 12 and 15 minutes. They ring together at 8:00 a.m. After how many minutes will they next ring together? easy

✅ Answer: (c) 180 minutes

💡 Explanation: The bells meet again after a time that is a multiple of all three intervals, so take the LCM. LCM(9, 12, 15) = 180 minutes, which is 11:00 a.m.

Q2. What is the largest container that can measure exactly 403 litres, 434 litres and 465 litres of milk? easy

✅ Answer: (a) 31 litres

💡 Explanation: The container must divide all three quantities, and it should be as large as possible, so take the HCF. 403 = 13 × 31, 434 = 14 × 31 and 465 = 15 × 31, so the HCF is 31.

Q3. What is the least number which leaves remainder 7 when divided by 18, 24 and 32? medium

✅ Answer: (b) 295

💡 Explanation: A number that leaves the same remainder r on division by several numbers is LCM + r. LCM(18, 24, 32) = 288, so the answer is 288 + 7 = 295.

Q4. The HCF of two numbers is 11 and their LCM is 693. One number is 77. What is the other? easy

✅ Answer: (d) 99

💡 Explanation: HCF × LCM equals the product of the two numbers: 11 × 693 = 7,623. Dividing by 77 gives 99.

Q5. What is the greatest number that divides 77, 147 and 252 leaving the same remainder in each case? medium

✅ Answer: (a) 35

💡 Explanation: If the remainders are equal, the differences 147 − 77 = 70 and 252 − 147 = 105 are exact multiples of the divisor. HCF(70, 105) = 35.

Q6. What is the smallest perfect square divisible by 6, 9 and 15? hard

✅ Answer: (c) 900

💡 Explanation: LCM(6, 9, 15) = 90 = 2 × 3² × 5. To make every prime's power even, multiply by 2 × 5. That gives 900 = 30².

How do I tell quickly whether a question needs HCF or LCM?

Ask whether the answer is smaller or larger than the numbers given. Cutting, packing and measuring into equal parts gives an answer no bigger than the numbers, which is HCF. Waiting for events to coincide gives an answer at least as big as the numbers, which is LCM.

Does HCF × LCM equal the product for three numbers too?

No. The rule HCF × LCM = product holds only for two numbers. For three or more numbers the product is usually larger than HCF × LCM, so do not apply the shortcut to them.

All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.

How this page was prepared. Drafted with AI assistance (Claude (Anthropic AI model)) from the sources listed above. 15 calculations on this page were recomputed by software and matched. Automated checks also covered structure, internal links and duplicate text. Software checks cannot confirm facts. Reviewed by praveenkumar on 2026-10-20 against the sources listed above.

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