The theory of numbers has been called a “neglected but singularly fascinating subject.” “Magic charm” is the quality ascribed to it by the foremost mathematician of the nineteenth century (Gauss).
Gauss said also: “Mathematics is the queen of the sciences, and the theory of number the crown of mathematics.” And he was master of the sciences of his time. He says “While it requires some facility in abstract reasoning, it may be taken up with practically no technical mathematics, is easily amenable to numerical exemplifications, and leads readily to the frontier. It is perhaps the only branch of mathematics where there is any possibility that new and valuable discoveries might be made without an extensive acquaintance with technical mathematics.”
The theory of numbers is a huge field of study of which the divisibility of numbers is an intimate part. Indeed, one of the key applications of the theory of numbers is deducing the divisibility of numbers.
Any number that divides another number is the factor of the number. The divisors of a number i.e., the numbers that divide a number without leaving a remainder may be determined by following the divisibility rules – rules for testing the divisibility of a number without actually performing division.
To understand the divisibility rules that govern the divisibility of numbers, it is imperative that we understand the notion of factors, as simply put, the numerical factors are the divisors of any number.
The results which arise from the multiplication of two or more numbers or letters are called products; and the numbers, or individual letters, are called factors. Factors of a number compose the number; and when taken separately, or in combination with other factors, will always divide our original number.
Factors are therefore fundamental in our study of numbers for no other reason other than this: they compose a number. If we want to gain a deeper understanding of numbers, it is of utmost importance that we study its factors.
In his book entitled ‘First Steps in Algebra’, G.A. Wentworth explores the definitions of factors and coefficients in a lucid and intuitive way.
Factors- When a number consists of the product of two or more numbers, each of these numbers is called a factor of the product.
The sign x is generally omitted between a figure and a letter or between letters; thus, instead of 63 x a x b, we write 63ab and instead of a x b x c, we write abc.
There are some numbers which it is easy to perceive whether they are divisors of a given number or not.
Divisibility by 2
A given number is divisible by 2, if the last digit is even; it is divisible by 4, if the two last digits are divisible by 4 ; it is divisible by 8, if the three last digits are divisible by 8 ; and, in general, it is divisible by 2″, if the n last digits are divisible by 2″.
Divisibility by 3
A number is divisible by 3, if the sum of the digits is divisible by 3 ; it may be divided by 6, if, beside this, the last digit is even ; it is divisible by 9, if the sum of the digits may be divided by 9.
Divisibility by 5
Every number that has the last digit O or 5, is divisible by 5.
Divisibility by 11
A number is divisible by 11, when the sum of the first, third, fifths, &c. digits is equal to the sum of the second, fourth, sixth, &c. digits.