Thanks a lot for the detailed explanations .
Few more things that has been confusing me are things like GCF and LCM of two numbers .
When we try to find the GCF or LCM of two numbers , expression or an equation , we are trying to simplify it right ?
Not exactly, although finding the GCF or LCM of two (or more) numbers can be very useful when simplifying. GCF stands for
Greatest
Common
Factor, and it is the largest number that is a factor of all of the original numbers. You may also see it written as GCD, with the D standing for
Divisor. LCM stands for
Least
Common
Multiple and is the smallest number that evenly divides all of the original numbers. You may also see this written as LCD, with the D standing for
Denominator.
Some examples: The GCF of 128 and 900 can be found by breaking each of them down into their prime factors. 128 = 2
7 and 900 = 2
2 * 3
2 * 5
2. The prime factors they share in common are 2
2, so their greatest common factor is 4. The GCF of 312, 900, and 1152 is 12, because 312 = 2
3 * 3 * 13, 900 = 2
2 * 3
2 * 5
2, and 1152 = 2
7 * 3
2. They share 2
2 and 3 as common factors, so their GCF is 2
2 * 3 = 12.
The LCM of 4 and 8 is 8, because 8 is the smallest number that evenly divides both 4 and 8. The LCM of 3 and 4 is 12, because 12 is the smallest number that evenly divides both 3 and 4. The LCM of two (or more) numbers will not always be equal to their product, but their product sets an upper bound. In other words, the LCM of some set of numbers will never be
more than their product. The LCM of 4, 6, and 8 is only 24, even though their product is 192.
One way to find the LCM of some numbers is to again write out their prime factorization. Returning to the example of 312, 900, and 1152, we see that the LCM is equal to all of their prime factorizations multiplied together. 1152 has 2
7 in its prime factorization. Out of the three numbers, this is the highest power of 2, so we know that will part of their LCM. Similarly, 3
2 is the highest shared power of 3, leaving 5
2 and 13 as non-shared powers. Thus, their LCM is 2
7 * 3
2 * 5
2 * 13 = 374,400.
One place using the LCM is incredibly useful is for finding a common denominator. Suppose you wanted to add 5/7 + 6/11. To do so, you'd need to find their LCD of 77 and convert the fractions to share that denominator, and then you could add straight across: 55/77 + 42/77 = 97/77. The LCM can also be used with polynomial expressions. If you needed to find \(\displaystyle \dfrac{1}{x^2+3x+2} -
\dfrac{1}{x^2+5x+6}\), you'd find their LCD. You can do this by first factoring both, to get \(\displaystyle \dfrac{1}{(x+1)(x+2)} - \dfrac{1}{(x+2)(x+3)}\). Thus, their LCD would be \(\displaystyle (x+1)(x+2)(x+3)\), and the answer to the problem would be \(\displaystyle \dfrac{1}{(x+1)(x+2)} - \dfrac{1}{(x+2)(x+3)} = \dfrac{2}{(x+1)(x+2)(x+3)}\).