LCM and HCF
Every number has two useful companions. The largest number that divides it exactly, along with another number, is called the highest common factor, and the smallest number that both of them divide exactly is called the lowest common multiple. These two ideas, the HCF and the LCM, sit at the base of a great deal of arithmetic. Without a common denominator no fractions can be added, without a common measure no two lengths can be compared, and without the LCM no two repeating events can be brought together. For the ADRE Grade IV examination the topic is taught at a friendly level, using small numbers and everyday situations such as bells that ring together, tiles that cover a floor exactly, and sweets that must be shared equally. The questions are short and the method is fixed, so a candidate who can find both the HCF and the LCM of two or three small numbers quickly and correctly will find this one of the easiest places to collect marks.
Basic Terms
- Factor: a number that divides another number exactly, leaving no remainder.
- Multiple: the result obtained when a number is multiplied by a counting number.
- Common factor: a factor that is shared by two or more given numbers.
- Common multiple: a multiple that is shared by two or more given numbers.
- HCF or GCD: the greatest number that divides each of the given numbers exactly.
- LCM: the smallest number that each of the given numbers divides exactly.
- Co-prime numbers: two numbers whose only common factor is one.
- Prime factorisation: writing a number as a product of prime numbers.
The HCF is never larger than the smallest of the given numbers, and the LCM is never smaller than the largest of them. This simple fact is often enough to strike out two or three wrong options at the very beginning of a question.
Finding the HCF by the Listing Method
The listing method is the easiest way to begin. Write down all the factors of each number, circle the factors that appear in every list, and then choose the largest of the common factors. For the numbers eighteen and twenty-four, the factors of eighteen are one, two, three, six, nine and eighteen, and the factors of twenty-four are one, two, three, four, six, eight, twelve and twenty-four. The common factors are one, two, three and six, and the greatest of them is six, so the HCF is six. For small numbers this method is quick and makes the meaning of the HCF very clear, which is why the Master Book: Maths Masti introduces the topic in this way before moving to shorter methods.
Finding the LCM by the Listing Method
The LCM is found in the same spirit by listing multiples. Write the multiples of each number in order and stop at the first multiple that appears in every list. For four and six, the multiples of four are four, eight, twelve, sixteen and so on, and the multiples of six are six, twelve, eighteen and so on. The first common multiple is twelve, so the LCM is twelve. A useful habit is to list the multiples of the larger number first, because its common multiple will be found sooner. Both the HCF and the LCM therefore become visible from the same small effort, and the link between factors and multiples becomes easy to see.
The Prime Factorisation Method
For slightly larger numbers, prime factorisation is the most reliable method. Break each number into its prime factors and write them as powers. For the HCF, take only the primes that appear in every number and give each one the smallest power it has in any of the lists. For the LCM, take every prime that appears in any number and give each one the largest power it has. For twelve and eighteen, the factorisations are two squared times three and two times three squared. The HCF takes two once and three once, giving six, while the LCM takes two squared and three squared, giving thirty-six. The two answers are found from the same factorisations, which saves time in the examination.
The Short Division or Ladder Method
When the numbers are a little bigger, the ladder method is the quickest of all. Write the numbers in a row and divide them by a common prime factor, carrying down any number that is not divisible. Repeat until no prime divides all the numbers together. The product of the divisors is the HCF, and the product of the divisors together with the numbers left in the last row is the LCM. This method presents both answers at the same time and is especially handy for three numbers, where it is much faster than listing. With a little practice the ladder becomes the candidate's favourite tool.
The Product Relation
For exactly two numbers there is a beautiful and very useful relation: the HCF multiplied by the LCM equals the product of the two numbers. Whenever a question gives the HCF, the LCM and one of the numbers, the other number can be found at once by dividing the product of the HCF and the LCM by the given number. The relation also serves as a check, because if the product of the HCF and the LCM does not match the product of the numbers, a mistake has been made somewhere. This rule holds only for two numbers and must never be used for three or more numbers, where the ladder or the factorisation method is required.
HCF and LCM of Fractions
Fractions have their own neat rule. The HCF of several fractions is the HCF of their numerators divided by the LCM of their denominators, and the LCM of several fractions is the LCM of their numerators divided by the HCF of their denominators. The rule appears in questions that ask for the largest fraction that divides several fractions exactly, or the smallest fraction that is a common multiple of several fractions. A candidate should memorise the rule with its two directions clearly separated, because the two cases are easily confused under examination pressure.
Everyday Word Problems
Word problems are the formulas dressed in a story, and Grade IV questions prefer these. Two bells that ring at intervals of, say, eight minutes and twelve minutes ring together again after a time equal to the LCM of the intervals, which is twenty-four minutes. The largest square tile that can cover a floor of given length and breadth has a side equal to the HCF of the two measurements. If sweets are to be packed equally into the largest possible number of boxes, the HCF decides the number, and if buses leaving at fixed intervals are to be counted until they meet again, the LCM decides the time. Whenever a question speaks of things meeting again, repeating together or fitting exactly, the LCM is at work; whenever it speaks of the greatest or the largest measure that fits, the HCF is at work.
How This Chapter Appears in the Examination
ADRE Grade IV, APSC, Assam Police and other state papers regularly set one or two questions from this chapter, and they are usually quick and friendly. The favourite forms are the product relation, the fraction rule, and the bell or tile story. Because the calculation is light and the method is fixed, this chapter rewards steady practice rather than cleverness, and a few sets of questions are normally enough to cover every variety that the paper can produce.
Common Mistakes to Avoid
- Using the product rule for three or more numbers, where it does not hold.
- Taking the highest power of a prime for the HCF, or the lowest power for the LCM.
- Mixing the two rules for fractions, using the HCF of the denominators where the LCM is needed.
- Continuing to divide in the ladder method after no prime divides all the numbers together.
- Believing that co-prime numbers must themselves be prime numbers.
How to Prepare This Topic
- Learn to list the factors and the multiples of small numbers confidently.
- Practise finding the HCF and the LCM by the listing method until the meaning is clear.
- Move on to prime factorisation, and then to the short division or ladder method.
- Memorise the product relation and the rule for fractions.
- Finish with everyday word problems about bells, tiles and shared sweets.
Frequently Asked Questions
What is the difference between the HCF and the LCM?
The HCF is the greatest number that divides the given numbers exactly, while the LCM is the smallest number that the given numbers divide exactly. The HCF is never bigger than the smallest given number, and the LCM is never smaller than the largest given number. For two numbers their product equals the product of the HCF and the LCM.
Is HCF multiplied by LCM always equal to the product of the numbers?
Only for exactly two numbers. For three or more numbers this relation does not hold, so the ladder method or prime factorisation must be used instead. This is one of the most common sources of error in the examination.
How do you find the LCM of fractions?
Divide the LCM of the numerators by the HCF of the denominators. The HCF of fractions is the reverse: the HCF of the numerators divided by the LCM of the denominators. Keeping the two directions apart is the whole trick.
Two bells ring at intervals of eight and twelve minutes. When do they ring together again?
They ring together after the LCM of eight and twelve, which is twenty-four minutes. The rule is that events repeating at fixed intervals repeat together after a time equal to the LCM of those intervals.
How should I prepare this topic for ADRE Grade IV?
Begin with the listing method so that the meaning of factors and multiples becomes clear, then learn prime factorisation, and after that the short division or ladder method. Practise the product relation, the rule for fractions, and the bell and tile word problems, and revise a small mixed set every week.