Answer :
To find the sum of the L.C.M (Least Common Multiple) and H.C.F (Highest Common Factor, also known as G.C.D) of the numbers 24, 36, and 48, we can follow these steps:
1. Find the H.C.F (G.C.D):
To calculate the H.C.F, we need to determine the largest number that divides all three numbers without leaving a remainder. We can do this by finding the common factors:
- The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
- The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
- The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.
The common factors are 1, 2, 3, 4, 6, and 12. The greatest of these is 12, so the H.C.F is 12.
2. Find the L.C.M (Least Common Multiple):
The L.C.M is the smallest number that all the numbers divide into without leaving a remainder. To find it, consider the highest power of each prime number that appears in the factorization of the given numbers:
- The prime factorization of 24 is [tex]\(2^3 \times 3^1\)[/tex].
- The prime factorization of 36 is [tex]\(2^2 \times 3^2\)[/tex].
- The prime factorization of 48 is [tex]\(2^4 \times 3^1\)[/tex].
The L.C.M will be the product of the highest powers of all prime numbers appearing in these factorizations:
- The highest power of 2 appearing is [tex]\(2^4\)[/tex].
- The highest power of 3 appearing is [tex]\(3^2\)[/tex].
Thus, the L.C.M = [tex]\(2^4 \times 3^2 = 16 \times 9 = 144\)[/tex].
3. Calculate the Sum of L.C.M and H.C.F:
Finally, add the H.C.F and the L.C.M:
[tex]\[
\text{Sum} = \text{H.C.F} + \text{L.C.M} = 12 + 144 = 156
\][/tex]
So, the sum of the L.C.M and H.C.F of 24, 36, and 48 is 156.
1. Find the H.C.F (G.C.D):
To calculate the H.C.F, we need to determine the largest number that divides all three numbers without leaving a remainder. We can do this by finding the common factors:
- The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
- The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
- The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.
The common factors are 1, 2, 3, 4, 6, and 12. The greatest of these is 12, so the H.C.F is 12.
2. Find the L.C.M (Least Common Multiple):
The L.C.M is the smallest number that all the numbers divide into without leaving a remainder. To find it, consider the highest power of each prime number that appears in the factorization of the given numbers:
- The prime factorization of 24 is [tex]\(2^3 \times 3^1\)[/tex].
- The prime factorization of 36 is [tex]\(2^2 \times 3^2\)[/tex].
- The prime factorization of 48 is [tex]\(2^4 \times 3^1\)[/tex].
The L.C.M will be the product of the highest powers of all prime numbers appearing in these factorizations:
- The highest power of 2 appearing is [tex]\(2^4\)[/tex].
- The highest power of 3 appearing is [tex]\(3^2\)[/tex].
Thus, the L.C.M = [tex]\(2^4 \times 3^2 = 16 \times 9 = 144\)[/tex].
3. Calculate the Sum of L.C.M and H.C.F:
Finally, add the H.C.F and the L.C.M:
[tex]\[
\text{Sum} = \text{H.C.F} + \text{L.C.M} = 12 + 144 = 156
\][/tex]
So, the sum of the L.C.M and H.C.F of 24, 36, and 48 is 156.
