Answer :
Of course! Let's break down the questions to provide a clear and detailed explanation.
### (i) What do you mean by the L.C.M. of these numbers?
The L.C.M. (Least Common Multiple) of two or more integers is the smallest positive integer that is divisible by each of the given integers without leaving a remainder.
For example, the LCM of 4 and 5 is 20, because 20 is the smallest number that both 4 and 5 can divide without any remainder.
In our case, we need to find the LCM of 25, 30, and 75.
### (ii) Find the least number which leaves no remainder when divided by the given numbers
To solve this, we need to find the LCM of 25, 30, and 75. We'll use the following steps:
1. Prime Factorization: First, factorize each number into its prime factors.
- [tex]\(25 = 5^2\)[/tex]
- [tex]\(30 = 2 \times 3 \times 5\)[/tex]
- [tex]\(75 = 3 \times 5^2\)[/tex]
2. Identify the highest power of each prime number:
- The highest power of 2 in the factorization is [tex]\(2^1\)[/tex] from 30.
- The highest power of 3 in the factorization is [tex]\(3^1\)[/tex] from 30 and 75.
- The highest power of 5 in the factorization is [tex]\(5^2\)[/tex] from 25 and 75.
3. Multiply these highest powers:
- [tex]\(2^1\)[/tex]
- [tex]\(3^1\)[/tex]
- [tex]\(5^2\)[/tex]
So, the LCM of 25, 30, and 75 is:
[tex]\[ \text{LCM} = 2^1 \times 3^1 \times 5^2 \][/tex]
4. Calculate the product:
[tex]\[ \begin{align*} 2^1 & = 2 \\ 3^1 & = 3 \\ 5^2 & = 25 \\ \end{align*} \][/tex]
Multiplying these together:
[tex]\[ 2 \times 3 \times 25 = 2 \times 3 = 6 \\ 6 \times 25 = 150 \][/tex]
Thus, the LCM of 25, 30, and 75 is 150.
Therefore, the least number which leaves no remainder when divided by 25, 30, and 75 is 150.
### (i) What do you mean by the L.C.M. of these numbers?
The L.C.M. (Least Common Multiple) of two or more integers is the smallest positive integer that is divisible by each of the given integers without leaving a remainder.
For example, the LCM of 4 and 5 is 20, because 20 is the smallest number that both 4 and 5 can divide without any remainder.
In our case, we need to find the LCM of 25, 30, and 75.
### (ii) Find the least number which leaves no remainder when divided by the given numbers
To solve this, we need to find the LCM of 25, 30, and 75. We'll use the following steps:
1. Prime Factorization: First, factorize each number into its prime factors.
- [tex]\(25 = 5^2\)[/tex]
- [tex]\(30 = 2 \times 3 \times 5\)[/tex]
- [tex]\(75 = 3 \times 5^2\)[/tex]
2. Identify the highest power of each prime number:
- The highest power of 2 in the factorization is [tex]\(2^1\)[/tex] from 30.
- The highest power of 3 in the factorization is [tex]\(3^1\)[/tex] from 30 and 75.
- The highest power of 5 in the factorization is [tex]\(5^2\)[/tex] from 25 and 75.
3. Multiply these highest powers:
- [tex]\(2^1\)[/tex]
- [tex]\(3^1\)[/tex]
- [tex]\(5^2\)[/tex]
So, the LCM of 25, 30, and 75 is:
[tex]\[ \text{LCM} = 2^1 \times 3^1 \times 5^2 \][/tex]
4. Calculate the product:
[tex]\[ \begin{align*} 2^1 & = 2 \\ 3^1 & = 3 \\ 5^2 & = 25 \\ \end{align*} \][/tex]
Multiplying these together:
[tex]\[ 2 \times 3 \times 25 = 2 \times 3 = 6 \\ 6 \times 25 = 150 \][/tex]
Thus, the LCM of 25, 30, and 75 is 150.
Therefore, the least number which leaves no remainder when divided by 25, 30, and 75 is 150.
