Finding Possible Bottle Sizes using Common Divisors

The problem asks us to find the number of different possible bottle sizes, in litres, that can be used to bottle 399 litres of petrol and 532 litres of diesel separately, with the bottle capacity being an integer. This means the bottle size must be a factor (or divisor) of 399 and also a factor of 532. In other words, the bottle size must be a common divisor of 399 and 532.

Identifying Common Divisors through GCD

The common divisors of two numbers are exactly the divisors of their Greatest Common Divisor (GCD). Therefore, to find all possible bottle sizes, we first need to calculate the GCD of 399 and 532. Once we find the GCD, the number of its divisors will give us the number of possible bottle sizes.

Calculating the GCD of 399 and 532

We can find the GCD by using prime factorization:

• Find the prime factors of 399:
• $399 = 3 \times 133$
• $133 = 7 \times 19$
• So, the prime factorization of 399 is $3 \times 7 \times 19$.

• Find the prime factors of 532:
• $532 = 2 \times 266$
• $266 = 2 \times 133$
• $133 = 7 \times 19$
• So, the prime factorization of 532 is $2^2 \times 7 \times 19$.

Now, let's find the common prime factors and their lowest powers:

• Common prime factors are 7 and 19.
• The lowest power of 7 is $7^1$.
• The lowest power of 19 is $19^1$.

The GCD is the product of these common prime factors raised to their lowest powers:

$\text{GCD}(399, 532) = 7^1 \times 19^1 = 7 \times 19 = 133$.

The GCD of 399 and 532 is 133.

Determining Possible Bottle Sizes (Divisors of GCD)

The possible bottle sizes are the divisors of the GCD, which is 133. Let's list the divisors of 133:

• 1 (Since 133 divided by 1 is 133)
• 7 (Since 133 divided by 7 is 19)
• 19 (Since 133 divided by 19 is 7)
• 133 (Since 133 divided by 133 is 1)

The divisors of 133 are 1, 7, 19, and 133.

Counting the Number of Possible Bottle Sizes

We found that the possible bottle sizes are 1 litre, 7 litres, 19 litres, and 133 litres. There are a total of 4 possible different bottle sizes.

Summary of Possible Bottle Sizes

Therefore, there are 4 different possible bottle sizes that can be used.

Revision Table: Key Concepts

Additional Information: Understanding GCD and Divisors

The concept of GCD is fundamental in number theory. The GCD of two numbers represents the largest possible size of a unit that can perfectly measure both quantities. In this problem, the GCD (133 litres) is the largest possible bottle size that can perfectly bottle both the petrol and the diesel without any leftovers.

Any smaller bottle size that can also perfectly bottle both quantities must also be a divisor of this largest possible size (the GCD). That's why the possible bottle sizes are the divisors of the GCD. If a number $d$ divides both $a$ and $b$, then $d$ must also divide $\text{GCD}(a, b)$. Conversely, every divisor of $\text{GCD}(a, b)$ divides both $a$ and $b$. This property is crucial for solving problems like this one.