Answer:
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Step-by-step explanation:
Let's break down the solution step-by-step:
### Part (i)
**The correct prime factorisation of 180 is:**
To factorize 180 into its prime factors, we repeatedly divide by the smallest prime number.
\[
180 \div 2 = 90 \\
90 \div 2 = 45 \\
45 \div 3 = 15 \\
15 \div 3 = 5 \\
5 \div 5 = 1
\]
So, the prime factorisation of 180 is:
\[
180 = 2^2 \times 3^2 \times 5
\]
### Part (ii)
**The LCM of two co-prime numbers is equal to:**
The Least Common Multiple (LCM) of two co-prime numbers is the product of those numbers. Co-prime numbers are numbers that have no common prime factors (other than 1).
### Part (iii)
**What will be the largest number of flowers he can join together without leaving a single flower:**
To determine the largest number of flowers he can use to form garlands such that no flowers are left out, we need to find the Greatest Common Divisor (GCD) of 200, 180, and 320.
First, let's find the prime factorisations:
- 200: \(2^3 \times 5^2\)
- 180: \(2^2 \times 3^2 \times 5\)
- 320: \(2^6 \times 5\)
The GCD is found by taking the lowest power of all common prime factors:
- The common prime factors are 2 and 5.
- The lowest power of 2 is \(2^2\).
- The lowest power of 5 is \(5^0\) (since 320 does not have 5 as a factor).
Thus, the GCD is \(2^2 = 4\).
So, the largest number of flowers he can join together without leaving a single flower is **4**.
### Part (iv)
**Find the LCM of 200 and 180:**
To find the LCM of 200 and 180, we take the highest power of all prime factors present in either number:
- For 2: the highest power is \(2^3\) (from 200).
- For 3: the highest power is \(3^2\) (from 180).
- For 5: the highest power is \(5^2\) (from 200).
So, the LCM is:
\[
LCM = 2^3 \times 3^2 \times 5^2
\]
Calculate it:
\[
2^3 = 8 \\
3^2 = 9 \\
5^2 = 25 \\
8 \times 9 \times 25 = 1800
\]
Therefore, the LCM of 200 and 180 is **1800**.
Answer:
(i)180=2*2*3*3*5
(ii)The LCM of two coprime number is equal to the product of the two numbers.
(iii)700flowers
(iv)1550
Step-by-step explanation:
