Some numbers are \( 28,36,54,72 \) and 144 . Vrite the prime factors of 28 . petermine the L.C.M. of the numbers using prime factors. Show that, the obtained L.C.M. by Euclid's process is equal to the obtained L.C.M. by prime factors

**Step 1. Write the prime factors of 28** The number \(28\) can be factored as follows: \[ 28 = 2 \times 14 = 2 \times 2 \times 7 = 2^2 \times 7 \] --- **Step 2. Determine the L.C.M. of \(28\), \(36\), \(54\), \(72\), and \(144\) using prime factors** 1. **Prime Factorization of Each Number:** - \(28 = 2^2 \times 7\) - \(36 = 2^2 \times 3^2\) - \(54 = 2 \times 3^3\) - \(72 = 2^3 \times 3^2\) - \(144 = 2^4 \times 3^2\) 2. **Choose the Highest Powers for Each Prime:** - For prime \(2\): the highest power is \(2^4\) (from \(144\)). - For prime \(3\): the highest power is \(3^3\) (from \(54\)). - For prime \(7\): the highest power is \(7^1\) (from \(28\)). 3. **Write the L.C.M.:** \[ \text{L.C.M.} = 2^4 \times 3^3 \times 7 \] 4. **Calculate the Value:** - \(2^4 = 16\) - \(3^3 = 27\) Now, compute: \[ 16 \times 27 = 432 \quad \text{and} \quad 432 \times 7 = 3024 \] Thus, the L.C.M. is \(\boxed{3024}\). --- **Step 3. Show that the L.C.M. by Euclid's Process Is Equal to the L.C.M. by Prime Factors** We can calculate the L.C.M. iteratively using the formula for two numbers: \[ \text{L.C.M.}(a, b) = \frac{a \times b}{\gcd(a, b)} \] 1. **L.C.M. of \(28\) and \(36\):** - First, find \(\gcd(28,36)\): \[ 36 \mod 28 = 8,\quad 28 \mod 8 = 4,\quad 8 \mod 4 = 0 \quad \Rightarrow \quad \gcd(28,36) = 4 \] - Then, compute: \[ \text{L.C.M.}(28,36) = \frac{28 \times 36}{4} = 28 \times 9 = 252 \] 2. **L.C.M. of \(252\) and \(54\):** - Find \(\gcd(252,54)\): \[ 252 \mod 54 = 36,\quad 54 \mod 36 = 18,\quad 36 \mod 18 = 0 \quad \Rightarrow \quad \gcd(252,54) = 18 \] - Then, compute: \[ \text{L.C.M.}(252,54) = \frac{252 \times 54}{18} = 14 \times 54 = 756 \] 3. **L.C.M. of \(756\) and \(72\):** - Find \(\gcd(756,72)\): \[ 756 \mod 72 = 36,\quad 72 \mod 36 = 0 \quad \Rightarrow \quad \gcd(756,72) = 36 \] - Then, compute: \[ \text{L.C.M.}(756,72) = \frac{756 \times 72}{36} = 756 \times 2 = 1512 \] 4. **L.C.M. of \(1512\) and \(144\):** - Find \(\gcd(1512,144)\): \[ 1512 \mod 144 = 72,\quad 144 \mod 72 = 0 \quad \Rightarrow \quad \gcd(1512,144) = 72 \] - Then, compute: \[ \text{L.C.M.}(1512,144) = \frac{1512 \times 144}{72} = 1512 \times 2 = 3024 \] The L.C.M. obtained by Euclid's process is \(\boxed{3024}\), which is the same as the L.C.M. obtained by using prime factors. --- \(\boxed{3024}\)