To find the smallest number of gallons of gas you need to buy for a whole number of dollars, you can set up the problem as follows:
Let \( x \) represent the number of gallons purchased. The total cost in dollars will be given by the equation:
\[ \text{Total cost} = 3.15x \]
We want \( 3.15x \) to be a whole number. This means that \( x \) must be chosen in such a way that \( 3.15x \) does not have a fractional part. To analyze this, we can express \( 3.15 \) as a fraction:
\[ 3.15 = \frac{315}{100} \]
Thus, the equation for total cost becomes:
\[ \text{Total cost} = \frac{315}{100}x \]
For \( \frac{315}{100}x \) to be a whole number, \( x \) must be chosen such that \( \frac{315x}{100} \) results in an integer. This requires \( 315x \) to be a multiple of \( 100 \).
Next, we need to find the smallest \( x \) so that \( 315x \) is divisible by \( 100 \). This means that we can find \( x \) by solving the equation:
\[ 315x \equiv 0 \pmod{100} \]
To simplify \( 315 \mod 100 \):
\[ 315 \mod 100 = 15 \]
We want \( 15x \equiv 0 \pmod{100} \). To solve for \( x \), \( x \) must make \( 15x \) a multiple of \( 100 \). The smallest \( x \) can be determined by finding the least common multiple (LCM) of \( 15 \) and \( 100 \).
The prime factorization is:
- \( 15 = 3 \times 5 \)
- \( 100 = 2^2 \times 5^2 \)
The LCM combines the highest power of each prime:
- For \( 2: 2^2 \)
- For \( 3: 3^1 \)
- For \( 5: 5^2 \)
Thus, the LCM is:
\[ \text{LCM}(15, 100) = 2^2 \times 3^1 \times 5^2 = 4 \times 3 \times 25 = 300 \]
Now, we need to find \( x \) such that \( 15x = 300 \):
\[ x = \frac{300}{15} = 20 \]
Therefore, the smallest number of gallons of gas you would have to buy for the total price to be a whole number of dollars is:
\[ \boxed{20} \]
1 answer