Question

b. The price displayed at a gas station is the price per gallon $3.146. 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 you need to buy for the total price to be a whole number of cents and a whole number of dollars, we can start by analyzing the price per gallon, which is $3.146.

### **Whole Number of Cents:**

1. The price per gallon is $3.146, which is equivalent to 314.6 cents.
2. To find when this would yield a whole number of cents, we need to find a number of gallons (let's say \( x \)) such that:
\[
314.6 \times x
\]
is a whole number.

3. Since 314.6 has a decimal in the tenths place, we can express it as a fraction:
\[
314.6 = \frac{3146}{10}
\]

4. Therefore, the total price in cents is:
\[
\frac{3146}{10} \times x = \frac{3146x}{10}
\]

5. For \( \frac{3146x}{10} \) to be a whole number, \( x \) must be such that \( 3146x \) is divisible by 10. Since \( 3146 \) itself is not divisible by 10, \( x \) must be a multiple of 10.

6. The smallest positive integer that is a multiple of 10 is \( 10 \).

**Result for whole number of cents:**
- You need to buy at least **10 gallons**.

### **Whole Number of Dollars:**

1. Now, for the price to be a whole number of dollars, we need:
\[
3.146 \times x
\]
to be a whole number.

2. In this case, we can express this as:
\[
3.146 \times x = \frac{3146}{100} \times x = \frac{3146x}{100}
\]

3. For \( \frac{3146x}{100} \) to be a whole number, \( 3146x \) must be divisible by 100. This requires \( x \) to be such that \( 3146x \) is a multiple of 100.

4. Since \( 3146 \mod 100 = 46 \), we can consider:
\[
46x \equiv 0 \mod 100
\]

5. To find the smallest \( x \) satisfying this, we need to find the least \( x \) such that \( 46x \) is a multiple of 100. This can be done using the smallest integer \( x \) satisfying:
\[
46x \equiv 0 \mod 100
\]

6. We can check multiples of 100:
- The least \( x \) can be calculated using the greatest common divisor (gcd) of 46 and 100, which is 2. Therefore:
\[
\frac{100}{\text{gcd}(46, 100)} = \frac{100}{2} = 50
\]

**Result for whole number of dollars:**
- You need to buy at least **50 gallons**.

### **Final Summary:**
- **Whole number of cents:** 10 gallons.
- **Whole number of dollars:** 50 gallons.