11 Plus LCM and HCF
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LCM and HCF are among the most commonly used topic in mathematics. This is because they form the fundamentals for solving different quantitative problems. LCM stands Least common multiples. HCF stands Highest Common Factor.
Questions on this topic appear in the 11 plus entrance exams conducted by 11 plus grammar schools, independent schools and other private schools. This topic is external linking to solving times table, time and speed, time and work-related problems.
We need to look into the buzz words Factors and Multiples for a clear understanding of these concepts. HCF and LCM depend on prime factorization.
The number that divides the given positive integer without leaving any remainder is called the factor.
For example, 24 ÷ 8 gives remainder=0.
Hence 8 is the factor of 24.
Multiple of a number is the result of multiplying a number by a whole number.
For example, the multiples of 4 can be obtained as
(4×1)=4, (4×2)=8,(4×3)=12,..... and so on.
Let us look at the definition of each acronym:
It is an acronym for “ Least Common Multiple”. It is the least number which is exactly divisible by each of the given numbers.
It is an acronym for “ Highest Common Factor “. It is the largest number that divides each of the given numbers. It is also called by the name GCD ( Greatest Common Divisor). It is calculated using factor trees.
The next question is how to compute them.
Finding Least Common Multiple LCM:
Given to find the LCM of 6 and 8, we proceed as follows:
List out the multiples of 6 and 8
The multiples of 6 are: 6×1=6, 6×2=12, 6×3=18, 6×4=24, 6×5=30,
The multiples of 8 are: 8×1=8, 8×2=16, 8×3=24, 8×4=32, 8×5=40,
The lowest and common multiple of both 6 and 8 is 24.
Hence LCM of 6 and 8 is 24.
Finding Highest Common Factor :
To compute the HCF of 6 and 12, we proceed as follows:
List out the factors of 6 and 12
The factors of 6 are: 1,2,3 and 6
The factors of 12 are: 1,2,3,4,6 and 12
The highest and common factor of both 6 and 12 is 6.
Hence LCM of 6 and 12 is 6.
Compute GCD and LCM using prime factors:
We employ this method when given numbers are large such that listing out all the factors and multiples for each number would be difficult.
Example problems of our website:
Properties of LCM and HCF:
- The HCF of given numbers is not greater or more than any of the numbers.
- The LCM of given numbers is not less than any of the numbers.
- The HCF of two or more prime numbers is always 1.
- The LCM of two or more prime numbers is the product of them.
- The given two numbers, ‘p' and ‘q’, are equal to their HCF and LCM product. This means:
p × q = ( LCM of p and q ) × ( HCF of p and q )
- LCM of any fraction = LCM of Numerators HCF of Denominators
- HCF of Fractions = HCF of NumeratorsLCM of Denominators
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