If we have two positive integers where one is greater than the other, which quantity would be greater (if possible): the larger integer divided by the smaller, or the result of dividing the least common multiple by the greatest common factor of the two integers? Since every number is both a divisor of itself and a multiple of itself, I know that the least common multiple could be the greater of the two integers at the least, and that the greatest common factor could be the lesser of the two integers at the most. Otherwise, the least common multiple at most could be the multiplication of the two integers divided by the greatest common factor. What would be the METHOD we would use to reason the relationship between the quantities?

When the product ( 5 1 7 ) ⋅ ( 2 3 ) is expressed as a mixed number, the answer is of the form a b/c.  What is a + b + c?

Ellie is putting together a balloon bouquet for her mom's birthday party. She wants to fill the bouquet with 48 balloons, and wants to make sure that the colors are perfectly divided into equal groups so that these are the same amount of every color. How many different ways can Ellie divide the balloons into equal groups such that each group contains the same number of balloons? The answer will require finding all the factors of 48.

A chemical manufacturer wants to lease a fleet of 24 railroad tank cars with a combined carrying capacity of 420,000 gallons. Tank cars with three different carrying capacities are available: 7,000 gallons, 14,000 gallons, and 28,000 gallons. How many of each type of tank car should be leased? Let x, be the number of cars with a 7,000 gallon capacity, x₂ be the number of cars with a 14,000 gallon capacity, and x3 be the number of cars with a 28,000 gallon capacity. Select the correct choice below and fill in the answer boxes within your choice. O A. The unique solution is x₁ = X₂ , and x3 = (Simplify your answers.) O B. There are multiple possible combinations of how the tank cars should be leased. The combinations are obtained from the equations x₁ = (Simplify your answers. Type integers or simplified fractions.) O C. There is no solution. t+( x₂ = t+ and x3 = t for sts