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In a store there is 345 litres rum, 120 ...

In a store there is 345 litres rum, 120 litres bear and 225 litres whiskey available. Find the greatest measure which can measure these completely.

A

25 liters

B

15 liters

C

35 liters

D

5 liters

Text Solution

AI Generated Solution

The correct Answer is:
To find the greatest measure that can completely measure 345 litres of rum, 120 litres of beer, and 225 litres of whiskey, we need to determine the Highest Common Factor (HCF) of these three quantities. Here’s how to solve it step by step: ### Step 1: Prime Factorization We will start by finding the prime factorization of each quantity. 1. **Prime Factorization of 345:** - 345 is divisible by 5 (since it ends in 5): \[ 345 \div 5 = 69 \] - Next, we factor 69: - 69 is divisible by 3: \[ 69 \div 3 = 23 \] - Therefore, the prime factorization of 345 is: \[ 345 = 5 \times 3 \times 23 \] 2. **Prime Factorization of 120:** - 120 is even, so we divide by 2: \[ 120 \div 2 = 60 \] - Continue factoring: \[ 60 \div 2 = 30 \] \[ 30 \div 2 = 15 \] \[ 15 \div 3 = 5 \] - Therefore, the prime factorization of 120 is: \[ 120 = 2^3 \times 3 \times 5 \] 3. **Prime Factorization of 225:** - 225 is divisible by 5: \[ 225 \div 5 = 45 \] - Next, we factor 45: - 45 is divisible by 3: \[ 45 \div 3 = 15 \] \[ 15 \div 3 = 5 \] - Therefore, the prime factorization of 225 is: \[ 225 = 3^2 \times 5^2 \] ### Step 2: Identify Common Factors Now, we will identify the common prime factors from the factorizations: - **345:** \(5^1 \times 3^1 \times 23^1\) - **120:** \(2^3 \times 3^1 \times 5^1\) - **225:** \(3^2 \times 5^2\) The common prime factors are 3 and 5. ### Step 3: Determine the Lowest Powers Next, we take the lowest power of each common factor: - For \(3\): The lowest power is \(3^1\). - For \(5\): The lowest power is \(5^1\). ### Step 4: Calculate the HCF Now, we can calculate the HCF by multiplying these common factors: \[ HCF = 3^1 \times 5^1 = 3 \times 5 = 15 \] ### Conclusion The greatest measure that can completely measure 345 litres of rum, 120 litres of beer, and 225 litres of whiskey is **15 litres**. ---
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