Question

b. The price displayed at a gas station is the price per gallon. What is the smallest
number of gallons of gas you would have to buy for the price of the gas to
have a whole number of cents? A whole number of dollars?

Answer 1

To determine the smallest number of gallons of gas needed for the price to be a whole number of cents and a whole number of dollars, we need to consider the price per gallon.

### Whole Number of Cents
A whole number of cents implies that the total cost must be a multiple of \(0.01\) dollars. If the price per gallon is denoted as \(P\) (in dollars), the total cost \(C\) for \(n\) gallons is given by:

\[
C = n \times P
\]

For \(C\) to be a whole number of cents:
1. \(C\) must equal \(n \times P\), which we want to be an integer multiple of \(0.01\).
2. Rearranging gives us \(C = (n \times P) \times 100\). For \(C\) to be a whole number, \(100 \times n \times P\) must be an integer.

This means that \(n \times P\) must be a multiple of \(0.01\).

To find the smallest \(n\):
- If the gas price \(P\) can be written as \(x.yz\) (where \(y\) and \(z\) are digits), it means \(P = \frac{a}{100}\) for some integer \(a\).
- To make \(nP\) a whole number, let’s set \(nP = k\), where \(k\) is an integer. This means \(n \times \frac{a}{100} = k \Rightarrow n \times a = 100k\).

The smallest integer \(n\) that satisfies this would be dependent on \(a\). A common case would be to take \(a=1\), where \(P = 0.01\):
- Here, \(n = 100\) leads to \(100 \times 0.01 = 1\), which is an integer in cents.

If \(P\) is any other fraction / decimal, you can generally find the smallest \(n\) such that \(n \times a\) is a multiple of \(100\).

### Whole Number of Dollars
A whole number of dollars implies that the total cost must be a whole number (integer). For this, we can write the same equation:

\[
C = n \times P
\]

For \(C\) to be a whole number of dollars, \(n \times P\) must be an integer. This does not require any additional conditions like cents, so it could be simply:

\[
nP \in \mathbb{Z}
\]

The smallest \(n\) would be the reciprocal of the price per gallon (if \(P\) is rational), meaning \(n\) must be a multiple of the denominator of \(P\).

### Conclusion
- For whole **cents**, the smallest \(n\) can typically be \(100\) or derived based on the gas price.
- For whole **dollars**, \(n\) would be determined by the price of gas \(P\).

Generally:
- If \(P = \$2.79,\ n=100\).
- If \(P\) is different, the specific value for \(n\) might shift, but generally, finding the least common multiple will help to discover the required dollars and cents values.