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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 & 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 & Geometry
Greek mathematicians tried to relate numbers & 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 & hexagonal numbers also. This is the way the Greeks thought about numbers & their properties.
Groupitizing or Mathematizing Number Patterns
We saw that the pattern of the triangle number 6 can be mathematically expressed as 1 + 2 + 3.
This can be done with many other patterns.
If we look at a 4 X 4 pattern along the diagonal it can be seen as a pattern 1 + 2 + 3 + 4 + 3 + 2 + 1. This also can be developed as a way to represent any square number in terms of addition.
This strategy is being called "groupitizing" by math educator Jo Boler.
It is a sophisticated version of subitizing with identifies a set directly as a number. Groupitizing identifies several sub-groups which can be subitized individually and constructed as an addition expression.
Armstrong Numbers
Mathematicians have identified another set of numbers which they call Armstrong numbers.
An Armstrong number is a number that is equal to the sum of its own digits raised to the power of the number of digits in the number.
For example, the number 153 is an Armstrong number, because 1^3 + 5^3 + 3^3 = 1 + 125 + 27 = 153.
There are only a handful of Armstrong numbers in any given range of numbers, and they become increasingly rare as the number of digits in the number increases.
Some other Armstrong numbers are 371,470,471, 1634 & 8208.
They are sometimes used in computer programming as a way of testing the efficiency and accuracy of algorithms that perform arithmetic operations on numbers.
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