To find 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, we need to find the least common multiple (LCM) of the numerator and denominator of the price per gallon.
The price per gallon is $2.359, which can be written as the fraction 2359/1000. To find the LCM of 2359 and 1000, we prime factorize each number:
2359 = 7 × 337
1000 = 2³ × 5³
The LCM is found by taking the highest power of each prime factor that appears in either factorization:
LCM = 2³ × 5³ × 7 × 337 = 140840
Therefore, to have a whole number of cents, you would have to buy at least 140840 gallons of gas.
To find the smallest number of gallons of gas you would have to buy for the price of the gas to have a whole number of dollars, we need to look at the denominator of the price per gallon, which is 1000.
Since there are 100 cents in a dollar, we need the denominator to be divisible by 100 to have a whole number of dollars. The smallest number divisible by 1000 and 100 is their LCM, which is 1000.
Therefore, to have a whole number of dollars, you would have to buy at least 1000 gallons of gas.
The price displayed 2.359 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? Explain.
1 answer