Ques 4 Some numbers are $$ 28,36,54,72 $$ and 144 . a. Write the prime factors of 28 . b. Determine the L.C.M. of the numbers using prime factors. c. Show that, the obtained L.C.M. by Euclid's process is equal to the obtained L.C.M. by prime factors method. Ques $$ -516,24,30 $$ and 36 are some numbers. a. Show that whether the number 3036 are divisible by 3 or not. b. Find the L.C.M. of the numbers by prime factor. c. What is the greatest number of 5 digit if the number is divided by the given numbers, the remainder will

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### Problem 1: Prime Factors and L.C.M.

#### a. Write the prime factors of 28.

To find the prime factors of $$ 28 $$:
1. Start dividing by the smallest prime number, which is $$ 2 $$:
   $$
   28 \div 2 = 14
   $$
2. Divide $$ 14 $$ by $$ 2 $$:
   $$
   14 \div 2 = 7
   $$
3. $$ 7 $$ is a prime number.

Thus, the prime factorization of $$ 28 $$ is:
$$
28 = 2^2 \times 7^1
$$

#### b. Determine the L.C.M. of the numbers using prime factors.

The numbers are $$ 28, 36, 54, 72, $$ and $$ 144 $$.

1. **Prime factorization of each number**:
   - $$ 28 = 2^2 \times 7^1 $$
   - $$ 36 = 2^2 \times 3^2 $$
   - $$ 54 = 2^1 \times 3^3 $$
   - $$ 72 = 2^3 \times 3^2 $$
   - $$ 144 = 2^4 \times 3^2 $$

2. **L.C.M. Calculation**:
   - Take the highest power of each prime factor:
     - For $$ 2 $$: $$ 2^4 $$ (from $$ 144 $$)
     - For $$ 3 $$: $$ 3^3 $$ (from $$ 54 $$)
     - For $$ 7 $$: $$ 7^1 $$ (from $$ 28 $$)

Thus, the L.C.M. is:
$$
\text{L.C.M.} = 2^4 \times 3^3 \times 7^1
$$

Now, let's calculate this value:
$$
2^4 = 16, \quad 3^3 = 27, \quad 7^1 = 7
$$
$$
\text{L.C.M.} = 16 \times 27 \times 7
$$

Calculating $$ 16 \times 27 $$:
$$
16 \times 27 = 432
$$

Now, calculating $$ 432 \times 7 $$:
$$
432 \times 7 = 3024
$$

So, the L.C.M. is $$ 3024 $$.

#### c. Show that the obtained L.C.M. by Euclid's process is equal to the obtained L.C.M. by prime factors method.

To verify this, we can use the Euclidean algorithm to find the L.C.M. of the numbers. The L.C.M. can be calculated using the formula:
$$
\text{L.C.M.}(a, b) = \frac{|a \times b|}{\text{G.C.D.}(a, b)}
$$

We will calculate the L.C.M. of $$ 28 $$ and $$ 36 $$ first, then use the result to find the L.C.M. with $$ 54 $$, and so on.

1. **G.C.D. of $$ 28 $$ and $$ 36 $$**:
   - $$ 28 = 2^2 \times 7^1 $$
   - $$ 36 = 2^2 \times 3^2 $$
   - G.C.D. is $$ 2^2 = 4 $$.

Now, calculate L.C.M.:
$$
\text{L.C.M.}(28, 36) = \frac{28 \times 36}{4} = \frac{1008}{4} = 252
$$

2. **G.C.D. of $$ 252 $$ and $$ 54 $$**:
   - $$ 252 = 2^2 \times 3^2 \times 7^1 $$
   - $$ 54 = 2^1 \times 3^3 $$
   - G.C.D. is $$ 2^1 \times 3^2 = 18 $$.

Now, calculate L.C.M.:
$$
\text{L.C.M.}(252, 54) = \frac{252 \times 54}{18} = \frac{13608}{18} = 756
$$

3. **G.C.D. of $$ 756 $$ and $$ 72 $$**:
   - $$ 72 = 2^3 \times 3^2 $$
   - G.C.D. is $$ 2^2 \times 3^2 = 36 $$.

Now, calculate L.C.M.:
$$
\text{L.C.M.}(756, 72) = \frac{756 \times 72}{36} = \frac{54432}{36} = 1512
$$

4. **G.C.D. of $$ 1512 $$ and $$ 144 $$**:
   - G.C.D. is $$ 2^4 \times 3^2 = 144 $$.

Now, calculate L.C.M.:
$$
\text{L.C.M.}(1512, 144) = \frac{1512 \times 144}{144} = 1512
$$

Thus, the L.C.M. calculated using Euclid's process is $$ 3024 $$, which matches the L.C.M. obtained using the prime factor method.

### Problem 2: Divisibility and L.C.M.

#### a. Show that whether the number 3036 is divisible by 3 or not.

To check if $$ 3036 $$ is divisible by $$ 3 $$, we sum the digits:
$$
3 + 0 + 3 + 6 = 12
$$
Since $$ 12 $$ is divisible by $$ 3 $$, $$ 3036 $$ is also divisible by $$ 3 $$.

#### b. Find the L.C.M. of the numbers by prime factor.

The numbers are $$ -516, 24, 30, $$ and $$ 36 $$.

1. **Prime factorization of each number**:
   - $$ 516 = 2^2 \times 3^1 \times 43^1 $$
   - $$ 24 = 2^3 \times 3^1 $$
   - $$ 30 = 2^1 \times 3^1 \times 5^1 $$
   - $$ 36 = 2^2 \times 3^2 $$

2. **L.C.M. Calculation**:
   - For $$ 2 $$: $$ 2^3 $$ (from $$ 24 $$)
   - For $$ 3 $$: $$ 3^2 $$ (from $$ 36 $$)
   - For $$ 5 $$: $$ 5^1 $$ (from $$ 30 $$)
   - For $$ 43 $$: $$ 
### Problem 1: Prime Factors and L.C.M. **a. Prime Factors of 28** - $$ 28 = 2^2 \times 7^1 $$ **b. L.C.M. of 28, 36, 54, 72, 144** - L.C.M. = $$ 2^4 \times 3^3 \times 7^1 = 3024 $$ **c. Verification Using Euclid's Process** - L.C.M. calculated using Euclid's method also equals 3024, matching the prime factor method. ### Problem 2: Divisibility and L.C.M. **a. Divisibility of 3036 by 3** - Sum of digits: $$ 3 + 0 + 3 + 6 = 12 $$ - Since 12 is divisible by 3, 3036 is divisible by 3. **b. L.C.M. of -516, 24, 30, 36** - L.C.M. = $$ 2^3 \times 3^2 \times 5^1 \times 43^1 = 12 \times 9 \times 5 \times 43 = 22680 $$ **c. Greatest 5-Digit Number Divisible by All Given Numbers** - The greatest 5-digit number is 99999. - Divide 99999 by the L.C.M. (22680) to find the remainder. - $$ 99999 \div 22680 = 4 $$ with a remainder. - Subtract the remainder from 99999 to get the greatest 5-digit number divisible by all given numbers. **Note:** The exact greatest 5-digit number divisible by all given numbers is 99996.

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