Properties of Number Sets

Like individual numbers, groups of numbers are also related by some properties. Many pairs of 2 numbers have properties. Let us see some examples.

1. Common Factors – 2 numbers, except under special cases, have common factors.

   1. For example, 6 & 8 have only 2 as a common factor.

   2. 24 & 36 have 2, 3, 4, 6 and 12 as common factors.

   3. 5 & 8 do not have a common factor

2. Lowest Common Factor – the LCF of any two numbers is always 1.

3. Highest Common Factor – If there are multiple factors, the highest among them is called HCF or Highest Common Factor. HCF is useful in quickly reducing fractions to their simplest term.

   1. For 24 & 36 the HCF is 12.

4. Common Multiples – Any set of 2 numbers, will have many common multiples.

   1. 2 & 7 have 14, 28, 56 as common multiples

5. Highest Common Multiple – There is no upper limit to the number of common multiples.

6. Lowest Common Multiple – The lowest common multiple is called LCM or Lowest Common Multiple. LCM is useful to do fraction additions, subtractions & divisions

   1. The LCM of 2 & 7 is 14

7. Co-primes (or Relatively Prime or Mutually Prime)

   1. If 2 numbers do not have any common factor other than 1, they are called co-prime. 3 & 10 are co-primes.

HCF & LCM

HCF & LCM is not a concept like the four basic operations which emerged from observing patterns around us. They are properties which are useful in simplifying computations. LCM is useful while adding unlike fractions. HCF could be useful in simplifying a fraction with large numbers in both the numerator and denominator.

Euclid gave an algorithm for quickly finding the GCF (which he called GCD - Greatest Common Denominator) of any two numbers. This must be one of the earliest algorithms in arithmetic. It also shows that the concept of GCD was known in Greek times.

Math syllabi in India give too much importance to HCF & LCM. This is possibly the result of focussing on methods of computation. Conceptually HCF & LCM are not very useful, except in computations given above.

There are contrived life situations where the answers could be seen to be either HCF or LCM. Here are some examples just for interest.

LCM Situation

There are 2 bus routes A & B operating a bus stop. Bus ‘A’ arrives every 30 minutes and Bus ‘B’ arrives every 40 minutes. If they arrive together at 8 a.m on a day, when is the next time when they will arrive together?

This problem can be easily solved by writing the various arrival times in a table, rather than using the idea of LCM & computing the time. What is important is to understand it as a situation where the result is the LCM.

HCF Situation

A sweet shop owner observes that people order sweets either in multiples of 12 or 16. All sweets are of the same size. What size of box should the owner order so that he can use the same box for both types of orders?

This problem also can be easily solved by writing the various box sizes in a table, rather than using the idea of HCF. What is important is to understand it as a situation where the result is the HCF.