The Geometric Representation of Greatest Common Factor

A factor is an integer that divides another integer. The number 6 is a factor of 12 since 6 divides 12. It is easy to see that 1, 2, 3, 4, and 12 are also factors of 12. Looking at the numbers in the tables below, we can see that some numbers have only 2 factors, 1 and itself. These numbers are called prime numbers.

Two or more numbers can have common factors. For example, let us consider the factors of 12 and 18.

Factors of 12: 1, 2, 3, 4, 6, 12

Factors of 18: 1, 2, 3, 6, 9, 18  » Read more

The Geometry of the Least Common Denominator

In mathematics, in putting together things, we must have a commonality; we must add objects that belong to the same set. We add 2 apples and 4 apples to get 6 apples. We do not add apples and oranges and come up with a single- kind-of-fruit-sum.

Apples

This is also true with numbers and measurements: we add, subtract, multiply or divide numbers that belong to the same set or measures with the same unit. We do not add binary numbers to decimal numbers and get a result without conversion.  We must either convert binary numbers to decimal, or vice versa and then perform addition or any other operations. Also, in finding the area of a rectangle with length 10 inches and width 5 centimeters, the answer must either be in square inches or in square centimeters. » Read more

Making Sense of Exponential Growth

Money on Chess Squares

A chessboard has 64 squares. If we are going to place 1 cent in the first square, 2 cents in the second square, 4 cents in the third square, and so on, how much money do we need to fill all the 64 squares?

From the pattern above, it is clear that the amount of money placed in each square is twice than that of the amount placed in the preceding square.  If we are going to number the squares from 1 through 64, then the amount of money needed to be placed in each square is shown in the table below.

As we can see from the table, the amount of money in the 64th square is 184,464,625,987,328,000.00. If we have indeed placed the money, the total money on our chess board is » Read more

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