Question

Find the L.C.M of the following numbers in which one number is the factor of other:
1.$5,20$
2.$6,18$
3.$12,48$
4.$9,45$
What do you observe in the result obtained?

Answer 1

Hint: First, we need to know about the LCM concept.
The least common multiple can be defined as the least number or small number with the given numbers exactly divisible.
Here first we need to find the all-prime factors of the given two or more than two numbers and find the least among them.
Prime factorization is the process of finding original numbers from multiplying the prime numbers.
Where the prime number is expressed as the number greater than one and which are not the product of any two smaller natural numbers.

Complete step by step answer:
Now we are going to find the prime factors for the given two numbers.
$(a)5,20$
Here we find the prime factors one by one.
The number $5$itself a prime number by the prime number definition.
The number $20$has the prime factors as $20 = 2 \times 2 \times 5$
LCM expressed as the product of the highest power of each factor involved among both the numbers.
The expression of the LCM process is
$
  5\left| \!{\underline {\,
  {5,20} \,}} \right. \\
  2\left| \!{\underline {\,
  {1,4} \,}} \right. \\
  2\left| \!{\underline {\,
  {1,2} \,}} \right. \\
  1\left| \!{\underline {\,
  {1,1} \,}} \right. \\
 $
Thus, the LCM of the numbers $5,20$is $5 \times 2 \times 2 = 20$
$(b)6,18$
Here we find the prime factors one by one.
The number $6$ has the prime factors as $6 = 2 \times 3$
The number $18$has the prime factors as $18 = 2 \times 3 \times 3$
LCM expressed as the product of the highest power of each factor involved among both the numbers.
The expression of the LCM process is
$
  3\left| \!{\underline {\,
  {6,18} \,}} \right. \\
  3\left| \!{\underline {\,
  {2,6} \,}} \right. \\
  2\left| \!{\underline {\,
  {2,2} \,}} \right. \\
  1\left| \!{\underline {\,
  {1,1} \,}} \right. \\
 $
Thus, the LCM of the numbers $6,18$is $3 \times 3 \times 2 = 18$
$(c)12,48$
Here we find the prime factors one by one.
The number $12$ has the prime factors as $12 = 2 \times 2 \times 3$
The number $48$has the prime factors as $48 = 2 \times 2 \times 3 \times 2 \times 2$
LCM expressed as the product of the highest power of each factor involved among both the numbers.
The expression of the LCM process is
$
  2\left| \!{\underline {\,
  {12,48} \,}} \right. \\
  2\left| \!{\underline {\,
  {6,24} \,}} \right. \\
  2\left| \!{\underline {\,
  {3,12} \,}} \right. \\
  2\left| \!{\underline {\,
  {3,6} \,}} \right. \\
  3\left| \!{\underline {\,
  {3,3} \,}} \right. \\
  1\left| \!{\underline {\,
  {1,1} \,}} \right. \\
 $
Thus, the LCM of the numbers $12,48$is $2 \times 2 \times 2 \times 2 \times 3 = 48$
$(d)9,45$
Here we find the prime factors one by one.
The number $9$ has the prime factors as $9 = 3 \times 3$
The number $45$has the prime factors as $45 = 5 \times 3 \times 3$
LCM expressed as the product of the highest power of each factor involved among both the numbers.
The expression of the LCM process is
$
  3\left| \!{\underline {\,
  {9,45} \,}} \right. \\
  3\left| \!{\underline {\,
  {3,15} \,}} \right. \\
  5\left| \!{\underline {\,
  {1,5} \,}} \right. \\
  1\left| \!{\underline {\,
  {1,1} \,}} \right. \\
 $
Thus, the LCM of the numbers $9,45$is $5 \times 3 \times 3 = 45$
Hence, we observe that the smallest number out of two given terms and its multiple gives the larger number.
We can conclude that the LCM of the given numbers is the larger number.
Hence the LCM is $20,18,48,45$ respectively.

Note:
GCD is the greatest common divisors. If the given question is to find the GCD then we have $(a)5,20$$(b)6,18$$(c)12,48$$(d)9,45$
The $\gcd (5,20) = 5$
Then$\gcd (6,18) = 6$, where $\gcd (6,18) = 2,3,6$are the common divisors but $6$is the greatest.
The $\gcd (12,48) = 12$
The $\gcd (9,45) = 9$
Where $5,6,12,9$ are the greatest common integers respectively to the given question.
Remember that prime numbers are the number only divisible by itself and one.
The non-prime numbers are known as the composite numbers.