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:
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The price per gallon is $3.146, which is equivalent to 314.6 cents.
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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.
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Since 314.6 has a decimal in the tenths place, we can express it as a fraction: \[ 314.6 = \frac{3146}{10} \]
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Therefore, the total price in cents is: \[ \frac{3146}{10} \times x = \frac{3146x}{10} \]
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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.
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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:
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Now, for the price to be a whole number of dollars, we need: \[ 3.146 \times x \] to be a whole number.
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In this case, we can express this as: \[ 3.146 \times x = \frac{3146}{100} \times x = \frac{3146x}{100} \]
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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.
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Since \( 3146 \mod 100 = 46 \), we can consider: \[ 46x \equiv 0 \mod 100 \]
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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: \[ 46x \equiv 0 \mod 100 \]
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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.