LCM and HCF for the Assam Police Constable Exam
The highest common factor and the lowest common multiple are two of the most dependable tools in the Elementary Mathematics section of the SLPRB Assam Police Constable (AB & UB) paper. Individually they are simple, and together they answer a surprising number of practical questions. The highest common factor is the largest number that divides each of the given numbers exactly, and the lowest common multiple is the smallest number that each of the given numbers divides exactly. In the Assam Police test these two ideas appear in questions on bells ringing together, on tiles and planks of the largest size, on buses leaving a terminal, on distributing articles equally, and on numbers that leave the same remainder when divided by several divisors. Almost all of these questions are solved in one or two lines once the candidate decides whether the situation calls for the highest common factor or the lowest common multiple. The method followed here is the one from the Master Book: Maths Masti (D. Hurina), which prefers prime factorisation for clarity and the division method for speed, and which always checks the answer against the given numbers.
Why LCM and HCF Are a Reliable Scoring Pair
The topic is scoring because the concepts are fixed and the calculations are short. There is no long reasoning to reconstruct; the candidate simply applies the correct method. The only real decision is whether the question is about dividing something into equal parts, which calls for the highest common factor, or about something that happens again and again at regular intervals, which calls for the lowest common multiple. Once this decision is made, the rest is routine, which is exactly what an objective paper rewards.
Terms You Should Know
- Factor: a number that divides another exactly, leaving no remainder.
- Multiple: a number obtained by multiplying a given number by a counting number.
- Highest common factor: the greatest number that divides all the given numbers exactly.
- Lowest common multiple: the smallest number that is a multiple of all the given numbers.
- Co-prime numbers: numbers whose highest common factor is one.
- Prime factorisation: writing a number as a product of prime numbers.
Two numbers that are co-prime have a highest common factor of one, and their lowest common multiple is simply their product. This fact settles many questions without any written work.
The Highest Common Factor and the Lowest Common Multiple
The highest common factor answers the question of the largest equal share, and the lowest common multiple answers the question of the earliest common meeting point. If several planks are to be cut into equal pieces of the greatest possible length, the answer is the highest common factor of their lengths. If several bells ring at different intervals and the question asks when they will ring together again, the answer is the lowest common multiple of the intervals. Learning to read the question for the word "largest equal" or the word "together again" is the whole skill of the chapter.
The Prime Factorisation Method
Write each number as a product of primes. The highest common factor is the product of the common primes taken to their lowest power. The lowest common multiple is the product of all the primes that appear, taken to their highest power. For the numbers twelve and eighteen, the factorisations are two squared times three and two times three squared, so the highest common factor is two times three, which is six, and the lowest common multiple is two squared times three squared, which is thirty-six. This method is the clearest and is the one to use when the numbers are small and easily factorised.
The Division or Ladder Method
For larger numbers, the division method is faster. For the highest common factor, divide the numbers by a common prime repeatedly until no common prime is left, and multiply the divisors. For the lowest common multiple, divide by any prime that divides at least one of the numbers, bring down the numbers that are not divisible, and continue until the last row consists of ones, then multiply the divisors. The division method avoids writing the full factorisation and is less prone to error when the numbers are many.
The Product Relation and When It Applies
For any two numbers, the product of the highest common factor and the lowest common multiple equals the product of the two numbers. This relation lets a candidate find one of the two when the other, along with the two numbers, is known. It is a favourite in objective questions because the answer follows without factorisation. It must, however, be used with care: the relation holds only for two numbers, not for three or more, and the commonest error is to apply it to a group of three numbers.
HCF and LCM of Fractions
The highest common factor of two or more fractions is the highest common factor of the numerators divided by the lowest common multiple of the denominators. The lowest common multiple of two or more fractions is the lowest common multiple of the numerators divided by the highest common factor of the denominators. These two rules answer questions in which the given quantities are fractions, such as cutting several tapes of fractional length into equal pieces. Writing the fractions in their lowest terms before applying the rules keeps the numbers small.
Everyday Word Problems Solved by LCM and HCF
Among the most common word problems are: the largest tile or plank that can measure several lengths exactly, solved by the highest common factor; the smallest number that can be divided exactly by several numbers, solved by the lowest common multiple; the greatest number that divides several numbers leaving the same remainder, solved by the highest common factor of the differences; and the number of times several bells or lights will act together in a given period, solved by the lowest common multiple combined with the total time. Recognising the setting makes the correct choice immediate.
Remainder Problems and the Same-Remainder Cases
The lowest common multiple is also used to find the smallest number that leaves a given remainder when divided by several divisors: find the lowest common multiple of the divisors and add the common remainder. When the remainders are different, the problem may have no neat answer unless the differences are equal, in which case the lowest common multiple is adjusted by subtracting the common difference. For the same-remainder case with the highest common factor, subtract the remainder from each number and take the highest common factor of the differences.
Shortcut Techniques for the Exam Hall
Decide first whether the question wants the highest common factor or the lowest common multiple by looking for the words "largest equal" or "together again". For two numbers, use the product relation to save time. For co-prime numbers, the lowest common multiple is the product and the highest common factor is one. In remainder problems, work with the differences. Check the answer by confirming that it divides, or is divisible by, each of the given numbers.
Common Traps and Errors
- Using the product relation for three or more numbers.
- Confusing the highest common factor with the lowest common multiple in a word problem.
- Taking the lowest common multiple of the numerators when finding the highest common factor of fractions.
- Writing a number as a product of factors that are not prime.
- Forgetting to bring down the numbers that are not divisible in the division method.
- Assuming the lowest common multiple is always the product of the numbers, even when they are not co-prime.
A Study Plan for This Chapter
- Learn the meaning of factor, multiple, highest common factor and lowest common multiple.
- Practise the prime factorisation method until it is automatic.
- Learn the division method for both quantities.
- Memorise the product relation and the rules for fractions.
- Finish with word problems and remainder problems, and revise the two key words each week.
Frequently Asked Questions
How do I know whether a question wants the HCF or the LCM?
Look for the words "largest equal", as in cutting planks of the greatest length, which calls for the highest common factor, or the words "together again", as in bells ringing together, which calls for the lowest common multiple.
When can I use the product relation?
The product of the highest common factor and the lowest common multiple equals the product of the two numbers, but this relation holds only for two numbers. It must not be applied to three or more numbers.
How do I find the HCF and LCM of fractions?
The highest common factor of fractions is the highest common factor of the numerators divided by the lowest common multiple of the denominators. The lowest common multiple of fractions is the lowest common multiple of the numerators divided by the highest common factor of the denominators.
What is the LCM of two co-prime numbers?
For two co-prime numbers the lowest common multiple is simply their product, because they share no common prime factor and so the highest common factor is one.
How do I find the smallest number that leaves the same remainder when divided by several numbers?
Find the lowest common multiple of the divisors and add the common remainder. If the remainders are not the same, check whether the differences are equal, and if they are, subtract the common difference from the lowest common multiple.