Properties of Numbers 2

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Divisibility by a number

If a given number can be arranged in an array, with each row containing X tokens and if no token are left, then the number is divisible by X

24 can be arranged as an array with each row having 3 or 4 or 6 or 8 tokens. Hence all of these are factors on 24.

Any number can be arranged as a line number, vertically &amp; horizontally. Hence every number is divisible by both 1 and the number itself.

Factors

A number which can divide another number without leaving a remainder, is called a factor of that number. One & the number itself are "trivial" factors of any number.

Most numbers have factors & many of them multiple factors. 4 & 6 are factors of 12.

Prime numbers have no other factors, other than 1 & the number itself.

Factors Mostly Occur in Pairs

The factors of any number occur in pairs. For 12, the factors are 1 & 12, 2 & 6 and 3 & 4. They are even in number.

But square numbers are exceptions to this rule. The square roots, which are also factors, are the same. For example factors of 16 as 1 & 16, 2 & 8 and 4 & 4. Here 4 is the square root of 16.

Hence the number of factors of a square number will be odd.

Multiples

Every number has multiples and that too an infinite number of them.

Arithmetic &amp; Geometry

Greek mathematicians tried to relate numbers &amp; their properties to geometrical figures. We will see a few examples.

Square Numbers

A number which can be arranged in a square array is a square. For example, 16 tokens can be arranged as 4X4 array. Hence it is a square. Later we will learn that the side of the square gives the square root of that number. In the above case 4 is the Square Root of 16.

1,4,9,16 etc are squares.

Look at these squares of numbers from 1 t0 9 - 1,4, 9, 16, 25, 36, 49, 64, 81. The digits in the unit's place 14966941 form a palindromic number!

Triangle Numbers

Numbers which can be arranged in a triangular array are called Triangle Numbers.



We can see that the top row contains 1 token and the subsequent rows have token which are 1 more than the row just above it. So each triangle number is the sum of all natural numbers up to the number which makes up the bottom most row.

In the above example the rightmost triangle number is 6 which is 1 + 2 + 3.

Similarly, we can talk about pentagonal &amp; hexagonal numbers also. This is the way the Greeks thought about numbers &amp; their properties.

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