Question

What is the GCF and LCM for \[30\] , \[35\] , \[36\] , \[42\] ?

Answer 1

Hint: Greatest common factor of all the given numbers is known as GCF . GCF is also called HCF or Highest common factor . Least Common Multiple or LCM is the smallest positive number that is a common multiple of two or more numbers .

Complete step-by-step answer:
We will obtain GCF and LCM using the Prime factorization method .
First we factorize the numbers into prime numbers . We will start dividing the number by \[2\] then one by one by all the prime numbers till we can not the number any further . Finally write the given number as a product of all the factors .
\[2\left| \!{\underline {\,
  {30} \,}} \right. \]
\[3\left| \!{\underline {\,
  {15} \,}} \right. \]
    \[5\]

\[5\left| \!{\underline {\,
  {35} \,}} \right. \]
 \[7\]

\[2\left| \!{\underline {\,
  {36} \,}} \right. \]
\[2\left| \!{\underline {\,
  {18} \,}} \right. \]
\[3\left| \!{\underline {\,
  9 \,}} \right. \]
   \[3\]

\[2\left| \!{\underline {\,
  {42} \,}} \right. \]
\[3\left| \!{\underline {\,
  {21} \,}} \right. \]
   \[7\]

Now we can write
\[30\] as a product of \[30 = \,2 \times 3 \times 5\]
\[35\] as a product of \[35\, = 5 \times 7\]
\[36\] as a product of \[36 = \,2 \times 2 \times 3 \times 3\]
\[42\] as a product of \[42\, = \,2 \times 3 \times 7\]
For GCF we obtain the product of the common factors of all the numbers .
So here we can see no common prime factor . But they all have a factor which is not prime .
That is \[1\] .
So \[1\] is the GCF of these numbers .
 For LCM we must list down all the prime factors found as many times as these numbers occur most often for any one given prime number and then multiply all of them .
So LCM is = \[2 \times 2 \times 3 \times 3 \times 5 \times 7\]
= \[1260\]
So \[1260\] is the LCM

Note: There are multiple ways to find GCF- Calculating from factorization method ,Calculating from Prime factorization method , Division method or ladder method. There are multiple ways to find LCM - Calculating multiples and finding the smallest common multiple , Calculating from Prime factors , Division method or ladder method . Here we will use a method which will provide both GCF and LCM by the same calculation .
Factor :
A number or algebraic expression that divides another number or expression evenly.
Common Factor :
Common factors are the factors of two or more numbers which are the same.
Multiple:
When we multiply a number by another number the result we get is called multiple of the first number.
Common Multiple :
Common multiples are the multiples of two or more numbers which are the same .
Example : \[20\] is multiple of \[4\] as well as \[5\] . \[4\] and \[5\] both are factors of \[20\] \[5\] is a common factor for \[20\] and \[35\].