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Euclid’s division algorithm, as the name suggests, has to do with divisibility of

integers. Stated simply, it says any positive integer acan be divided by another positive

integer bin such a way that it leaves a remainder rthat is smaller than b. Many of you

probably recognise this as the usual long division process. Although this result is quite

easy to state and understand, it has many applications related to the divisibility properties

of integers. We touch upon a few of them, and use it mainly to compute the HCF of

two positive integers.

1.2 Euclid’s Division Lemma

Consider the following folk puzzle.

A trader was moving along a road selling eggs. An idler who didn’t have

much work to do, started to get the trader into a wordy duel. This grew into a

fight, he pulled the basket with eggs and dashed it on the floor. The eggs broke.

The trader requested the Panchayat to ask the idler to pay for the broken eggs.

The Panchayat asked the trader how many eggs were broken. He gave the

following response:

If counted in pairs, one will remain;

If counted in threes, two will remain;

If counted in fours, three will remain;

If counted in fives, four will remain;

If counted in sixes, five will remain;

If counted in sevens, nothing will remain;

My basket cannot accomodate more than 150 eggs.

An algorithmis a series of well defined steps

which gives a procedure for solving a type of

problem.

9

An equivalent version of Theorem 1.2 was probably first

recorded as Proposition 14 of Book IX in Euclid’s

Elements, before it came to be known as the Fundamental

Theorem of Arithmetic. However, the first correct proof

was given by Carl Friedrich Gauss in his Disquisitiones

Arithmeticae.

Carl Friedrich Gauss is often referred to as the ‘Prince of

Mathematicians’ and is considered one of the three

greatest mathematicians of all time, along with Archimedes

and Newton. He has made fundamental contributions to

both mathematics and science.

The Fundamental Theorem of Arithmetic says that every composite number

can be factorised as a product of primes. Actually it says more. It says that given

any composite number it can be factorised as a product of prime numbers in a

‘unique’ way, except for the order in which the primes occur. That is, given any

composite number there is one and only one way to write it as a product of primes,

as long as we are not particular about the order in which the primes occur. So, for

example, we regard 2 × 3 × 5 × 7 as the same as 3 × 5 × 7 × 2, or any other

possible order in which these primes are written. This fact is also stated in the

following form:

The prime factorisation of a natural number is unique, except for the order

of its factors.

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