Question

The numbers A, B, and C, written as a product of their prime factors are given below:

A= 22 3452133
B= 24 365275
C= 172 375273

Find:
The greatest whole number that is a factor of A,B and C
The smallest whole number that is divisible by all of A, B and C

Answer 1

Answer:

ASSUMED KNOWLEDGE

Fluency in multiplication and division are essential.

Divisibility tests, particularly by 2, 3, 5 and 11, are useful.

Index notation with whole-number indices, and square roots, are required.

The Highest Common Factor (HCF) and The Lowest Common Multiple (LCM), and cube and possibly higher roots, are required for the last two content items, which are

often studied a little later than prime factorisation.

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MOTIVATION

A fundamental technique in mathematics is to break something down into its component parts, and rebuild it from those parts. Thus we can factor any whole number into a product of prime numbers, for example

60 = 22 × 3 × 5

and this prime factorisation is unique, apart from the order of the factors. Conversely, if we are given the prime factors of a number, we can reconstruct the original whole number by multiplying the prime factors together,

(2 × 2) × (3 × 5) = 4 × 15 = 60 or (2 × 5) × (2 × 3) = 10 × 6 = 60

and we will always get the same original number, whatever order we choose for multiplying the prime factors.

In other situations, however, such processes do not work nearly as straightforwardly, as can be illustrated using the analogy of chemistry. Every compound can be broken down uniquely into its elements, but if we are given the elements, there are often a great many different compounds that can be formed from them.

Prime factorisation is a very useful tool when working with whole numbers, and will be used in mental arithmetic, in fractions, for finding square roots, and in calculating the HCF and LCM.

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CONTENT

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THE DEFINITION OF PRIME NUMBERS