Question

I. Find the H.C.F of (a) 16 and 9
                                             (b) 48, 84, 88
II. Find the L.C.M of (a) 16 and 12
                                  (b) 34, 85, 88

Answer 1

Hint: I. Find the factors of the numbers. To find the H.C.F (highest common factor) we need to first find its prime factors. prime factors are factors consisting of prime numbers. After finding the prime number of the two or more numbers separately we will see which factors are common between these numbers. thus giving us the highest common factor.
 II. Find the multiples of the numbers. First, we will make a table and divide all the numbers given to us accordingly with prime numbers. The numbers from which we have divided these numbers are the multiples here.finding the L.C.M means to find all the multiples between the given numbers. But, the multiples common between the given numbers should only be written once.


Complete step by step solution:
 1st part: (a)
List the prime factors of the numbers given.
  \[\begin{array}{*{35}{l}}
   \text{Factors of }16\text{= 2 x 2 x 2 x 2 x 1} \\
   \text{Factors of }9\text{ = 3 x 3 x 1} \\
\end{array}\]
Find the common factors.
 There is only 1 common factor.
Product of the common factors is the H.C.F
Hence, the H.C.F of 16 and 9 is 1.

1st part: (b)
List the prime factors of the numbers given.
\[\begin{array}{*{35}{l}}
   \text{Factors of 48 = 3 x 2 x 2 x 2 x 2 x 1} \\
   \begin{gathered}
  & \text{Factors of 84 = 7 x 3 x 2 x 2 x 1} \\
 & \text{Factors of 88 = 11 x 2 x 2 x 2 x 1} \\
\end{gathered} \\
\end{array}\]
Find the common factors.
The common factors =1, 2 and 2.
Product of the common factors is the H.C.F
      Hence, HCF of 48, 84 and 88 = 1x 2 x 2 = 4.






2nd part: (a)
Construct a table and arrange the given numbers as shown

Find the common multiples.

The common multiples = 2,2,2,2,3.

Product of the common multiples is the L.C.M

  Hence, L.C.M of 16 and 12 is 48.

2nd part: (b)

 Construct a table and arrange the given numbers as shown.

Find the common multiples.

The common multiples = 2,2,2,5,11,17.

Product of the common multiples is the L.C.M

      Hence, L.C.M of 34, 85 and 88 is 7480.

Additional Information: H.C.F stands for Highest Common Factor while L.C.M stands for Lowest Common Multiple. Here, we use the prime numbers for factorization. 1 is not a prime number nor a composite number.

Note: For calculating the L.C.M. and H.C.F. We have to use prime numbers for factorization. prime numbers are the numbers that have only two factors. The factor 1 and itself. Thus having prime numbers for factorization will give us all the factors and multiples without missing any single factors. Using composite numbers for factorization can lead to missing some factors and multiple common in between the numbers.