Explaining your method fully, determine whether or not 1189 is a prime number.

(i)     State the fundamental theorem of arithmetic.

(ii)     The positive integers M and N have greatest common divisor G and least common multiple L . Show that GL = MN .

any clearly indicated method of dividing 1189 by successive numbers     M1

find that 1189 has factors 29 and/or 41     A2

it follows that 1189 is not a prime number     A1

Note: If no method is indicated, award A1 for the factors and A1 for the conclusion.

[4 marks]

(i)     every positive integer, greater than 1, is either prime or can be expressed uniquely as a product of primes     A1A1

Note: Award A1 for “product of primes” and A1 for “uniquely”.

(ii)     METHOD 1

let M and N be expressed as a product of primes as follows

M = AB and N = AC     M1A1

where A denotes the factors which are common and B, C the disjoint factors which are not common

it follows that G = A     A1

and L = GBC     A1

from these equations, it follows that

\(GL = A \times ABC = MN\)     AG

METHOD 2

Let \(M = {2^{{x_1}}} \times {3^{{x_2}}} \times ...p_n^{{x_n}}\) and \(N = {2^{{y_1}}} \times {3^{{y_2}}} \times ...p_n^{{y_n}}\) where \({p_n}\) denotes the \({n^{{\text{th}}}}\) prime     M1

Then \(G = {2^{\min ({x_1},{y_1})}} \times {3^{\min ({x_2},{y_2})}} \times ...p_n^{\min ({x_n},{y_n})}\)     A1

and \(L = {2^{\max ({x_1},{y_1})}} \times {3^{\max ({x_2},{y_2})}} \times ...p_n^{\max ({x_n},{y_n})}\)     A1

It follows that \(GL = {2^{{x_1}}} \times {2^{{y_1}}} \times {3^{{x_2}}} \times {3^{{y_2}}} \times ... \times p_n^{{x_n}} \times p_n^{{y_n}}\)     A1

= MN     AG

[6 marks]

In (a), some candidates tried to use Fermat’s little theorem to determine whether or not 1189 is prime but this method will not always work and in any case the amount of computation involved can be excessive. For this reason, it is strongly recommended that this method should not be used in examinations. In (b), it was clear from the scripts that candidates who had covered this material were generally successful and those who had not previously seen the result were usually unable to proceed.

In (a), some candidates tried to use Fermat’s little theorem to determine whether or not 1189 is prime but this method will not always work and in any case the amount of computation involved can be excessive. For this reason, it is strongly recommended that this method should not be used in examinations. In (b), it was clear from the scripts that candidates who had covered this material were generally successful and those who had not previously seen the result were usually unable to proceed.