HCF & LCM - Basics 2

Wednesday, June 07, 2006

I assume the concepts of factors and multiples would be clear till now, if not, don't blame me for it ;), go through my previous posts, I am a bad bad teacher.

Least Common Multiple (LCM): What are the factors of 10? 2 & 5, correct. What if I ask you about a number which is divisible by both 2 and 5, what will be your answer? 10, correct. So, basically 10 is a common multiple of 2 and 5. Now what is a common multiple of 2, 3 & 5? In other words a number which is divisible by 2, 3 and 5? 30, correct. If you understand this concept then you already know what LCM is. Let's officially define LCM. LCM of two or more given numbers is the least number which is exactly divisible by each of the given numbers. Official definitions are scary, aren't they? Of course unless you are a nerd.

Q) Why should I learn LCM? Can I do without it?
A) Previously while we were looking into HCF, we planted some trees, now that the trees are grown and its Mango season, I want to send some Mangoes (hafus) to my siblings and relatives, I sent each one same amount of mangoes, dint exceed 150 dozens each, just that the cartoons didn't contain equal number of mangoes. I sent few cartoons each containing 6 dozens to my sister, I shipped few cartoons each containing 8 dozens to my brother and I shipped few cartoons each containing 10 dozens of mangoes to my aunty, who stays in Alibaug. How many cartoons were in each shipment?

Since I am fair to everyone, each receives equal number of mangoes, say X, then X has to be a multiple of 6, 8 and 10. In other words X is a common multiple of 6, 8 and 10. To find out common multiple which is not greater than 150, I did some math, you don't need to worry about that right now.

6 = 2 x 3
8 = 2^3
10 = 2 x 5

LCM = 2^3 x 3 x 5 = 120.

Each one received 120 dozens of mangoes, that's hell lot of mangoes. Hence my sister received 120 / 6 = 20 cartoons, my brother received 120 / 8 = 14 cartoons and my aunt who is in Alibaug received 120 / 10 = 12 cartoons, so total 20 + 14 + 12 = 46 cartoons.

So can I do without LCM? No. LCM also comes handy for arithmetic of fractions, which I will cover later.

Methods to find LCM:

Factorization method 1: Express the given numbers in prime factors, then pick all the highest powers of the primes and multiply it. Voila! you got your LCM.

Example: LCM of 60 and 150
60 = 2² x 3 x 5
150 = 2 x 3 x 5²
LCM = 2² x 3 x 5²

In the above example the highest powers of 2, 3 and 5 are 2, 1 and 2 respectively, hence I picked 2², 3 & 5².

Factorization method 2: Start with a small prime, divide all the numbers possible, leave alone which are not divisible, repeat until you are left with only prime numbers.

Example: Find the LCM of 26, 18, 28, 39

LCM = 2 x 2 x 3 x 3 x 7 x 13 = 3276

*TIP: After practice you will be able to find out the common factors by just looking at the numbers, only consider the highest power of that factor and continue.

Dont's: Don't mistake LCM for HCF, if you are finding the lingo little confusing, try to say LCM as "common multiple which is least" for LCM and "common factor which is highest" for HCF. You will get used to it.