You may be asked to calculate a least common multiple (LCM) or a highest common factor (HCF) in every other exam, either as part of a question, or as a question in itself. Use the quickest methods available to save yourself time.
lets start with basic questions to understand its meaning !
The Least Common Multiple (LCM) of two or more integers is the smallest integer that is a multiple of each of them. For example, the LCM of 5 and 3 is 15, and the LCM of 22 and 3 is 66.
The Greatest Common Factor (HCF) of two or more integers is the greatest integer that is a factor of each of the original numbers. For example, the HCF of 12 and 9 is 3, because the prime factors of 12 are 3 and 2, and the factor of 9 is 3, and so 3 is the largest common factor.
Least Common Multiple
Consider the following question:
What is the LCM of 91 and 98?
The key is to find the prime factors of each number. To make things easier, start with larger factors of the number, and keep breaking them down to the primes:
91: 13 * 7
98: 49 * 2 = 7 * 7 * 2
Take the highest count of each prime number.
There are two 7s in 98, but only one 7 in 91, so take two 7s, 2 and 13.
Multiply these together: 7 * 7 * 2 * 13 = 1274.
Therefore, the LCM of 91 and 98 is 1274.
Greatest Common Factor
Consider this question:
What is the greatest common factor of 84 and 144?
Split the numbers into their prime factors. And again, to make things easier, start with larger factors of the number, and keep breaking them down:
84: 7 * 12 = 7 * 3 * 4 = 7 * 3 * 2 * 2
144: 12 * 12 = 3 * 4 * 3 * 4 = 3 * 3 * 2 * 2 * 2 * 2
Take the prime factors that are common to both numbers.
Those are 3, 2 and 2.
Multiply these together: 3 * 2 * 2 = 12
Therefore, the HCF of 84 and 144 is 12.
When things get trickier
now lets focus on a medium level difficulty question, and how to solve quicker .
Consider the following question .
What is the lowest positive integer that is divisible by both 5,000,000 and 256?
The first thing you need to realize is that this is a thinly disguised request for you to calculate the LCM, since the LCM would be divisible by both 5,000,000 and 256.
Find the prime factors.
5,000,000: 5 * 1,000,000=
= 5 * 1000 * 1000
= 5 * 2 * 500 * 2 * 500
= 5 * 2 * 2 * 250 * 2 * 2 * 250
= 5 * 2 * 2 * 25 * 10 * 2 * 2 * 25 * 10
= 57 * 26
256: 28
Take the highest count of each prime number.
28 and 57.
Multiply these together: 28 * 57 = 27 * 57 * 2 = 107 * 2 = 20,000,000.
Therefore, the LCM of 5,000,000 and 256 is 20,000,000.
Follow these simple steps to make even questions involving large numbers succumb to your charms. Good luck!
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