91Ó°ÊÓ

(a) Factorise 12345 as a product of primes. (b) Using only mental arithmetic, make a list of all prime numbers up to 100 . (c)(i) Find a prime number which is one less than a square. (ii) Find another such prime. There are 4 prime numbers less than \(10 ; 25\) prime numbers less than \(100 ;\) and 168 prime numbers less than 1000 . Problem \(\mathbf{4}(\mathrm{c})\) is included to emphasise a frequently neglected message: Words and images are part of the way we communicate. But most of us cannot calculate with words and images. To make use of mathematics, we must routinely translate words into symbols. For example, unknown numbers need to be represented by symbols, and points in a geometric diagram need to be properly labelled, before we can begin to calculate, and to reason, effectively.

(a)(i) Explain why any integer that is a factor (or a divisor) of both \(m\) and \(n\) must also be a factor of their difference \(m-n,\) and of their sum \(m+n\). (ii) Prove that $$ H C F(m, n)=H C F(m-n, n) $$ (iii) Use this to calculate in your head \(H C F(1001,91)\) without factorising either number. (b)(i) Prove that: \(H C F(m, m+1)=1\). (ii) Find \(H C F(m, 2 m+1)\). (iii) Find \(H C F\left(m^{2}+1, m-1\right)\).

(a) Which of the prime numbers \(<100\) can be written as the sum of two squares? (b) Find an easy way to immediately write \(\left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right)\) in the form \(\left(x^{2}+y^{2}\right)\). (This shows that the set of integers which can be written as the sum of two squares is "closed" under multiplication.) (c) Prove that no integer (and hence no prime number) of the form \(4 k+3\) can be written as the sum of two squares. (d) The only even prime number can clearly be written as a sum of two squares: \(2=1^{2}+1^{2}\). Euler \((1707-1783)\) proved that every odd prime number of the form \(4 k+1\) can be written as the sum of two squares in exactly one way. Find all integers \(<100\) that can be written as a sum of two squares. (e) For which integers \(N<100\) is it possible to construct a square of area \(N\), with vertices having integer coordinates? In Problem 25 parts (a) and (d) you had to decide which integers \(<100\) can be written as a sum of two squares as an exercise in mental arithmetic. In part (b) the fact that this set of integers is closed under multiplication turned out to be an application of the arithmetic of norms for complex numbers. Part (e) then interpreted sums of two squares geometrically by using Pythagoras' Theorem on the square lattice. These exercises are worth engaging in for their own sake. But it may also be of interest to know that writing an integer as a sum of two squares is a serious mathematical question \- and in more than one sense. Gauss \((1777-1855),\) in his book Disquisitiones arithmeticae (1801) gave a complete analysis of when an integer can be represented by a 'quadratic form', such as \(x^{2}+y^{2}\) (as in Problem 25) or \(x^{2}-2 y^{2}\) (as in Problem \(\mathbf{5 4}(\mathrm{c})\) in Chapter 2 ). A completely separate question (often attributed to Edward Waring \((1736-1798))\) concerns which integers can be expressed as a \(k^{\text {th }}\) power, or as a sum of \(n\) such powers. If we restrict to the case \(k=2\) (i.e. squares), then: \- When \(n=2,\) Euler \((1707-1783)\) proved that the integers that can be written as a sum of two squares are precisely those of the form $$ m^{2} \times p_{0} \times p_{1} \times p_{2} \times \cdots \times p_{s} $$ where \(p_{0}=1\) or \(2,\) and \(p_{1}

Problem 27 (Overlapping squares) A square \(A B C D\) of side 2 sits on top of a square \(P Q R S\) of side \(1,\) with vertex \(A\) at the centre \(O\) of the small square, side \(A B\) cutting the side \(P Q\) at the point \(X,\) and \(\angle A X Q=\theta\) (a) Calculate the area of the overlapping region. (b) Replace the two squares in part (a) with two equilateral triangles. Can you find the area of overlap in that case? What if we replace the squares (i.e. regular 4-gons) in part (a) with regular \(2 n\) -gons? \(\Delta\)

Problem 28 (A folded triangle) The equilateral triangle \(\triangle A B C\) has sides of length \(1 \mathrm{~cm} . \quad D\) and \(E\) are points on the sides \(A B\) and \(A C\) respectively, such that folding \(\triangle A D E\) along \(D E\) folds the point \(A\) onto \(A^{\prime}\) which lies outside \(\triangle A B C\) What is the total perimeter of the region formed by the three single layered parts of the folded triangle (i.e. excluding the quadrilateral with a folded layer on top )?

(a) Joining the midpoints of the edges of an equilateral triangle \(A B C\) cuts the triangle into four identical smaller equilateral triangles. Removing one of the three outer small triangles (say \(A M N\), with \(M\) on \(A C\) ) leaves three-quarters of the original shape in the form of an isosceles trapezium \(M N B C .\) Show how to cut this isosceles trapezium into four congruent pieces. (b) Joining the midpoints of opposite sides of a square cuts the square into four congruent smaller squares. If we remove one of these squares, we are left with three-quarters of the original square in the form of an L-shape. Show how to cut this L-shape into four congruent pieces.

Imagine a triangle \(A B C\) on the unit sphere (with radius \(r=\) 1), with angle \(\alpha\) between \(A B\) and \(A C,\) angle \(\beta\) between \(B C\) and \(B A,\) and angle \(\gamma\) between \(C A\) and \(C B\). You are now in a position to derive the remarkable formula for the area of such a spherical triangle. (a) Let the two great circles containing the sides \(A B\) and \(A C\) meet again at \(A^{\prime} .\) If we imagine \(A\) as being at the North pole, then \(A^{\prime}\) will be at the South pole, and the angle between the two great circles at \(A^{\prime}\) will also be \(\alpha .\) The slice contained between these two great circles is called a lune with angle \(\alpha\) (i) What fraction of the surface area of the whole sphere is contained in this lune of angle \(\alpha ?\) Write an expression for the actual area of this lune. (ii) If the sides \(A B\) and \(A C\) are extended backwards through \(A,\) these backward extensions define another lune with the same angle \(\alpha,\) and the same surface area. Write down the total area of these two lunes with angle \(\alpha\) (b)(i) Repeat part (a) for the two sides \(B A, B C\) meeting at the vertex \(B,\) to find the total area of the two lunes meeting at \(B\) and \(B^{\prime}\) with angle \(\beta\). (ii) Do the same for the two sides \(C A, C B\) meeting at the vertex \(C,\) to find the total area of the two lunes meeting at \(C\) and \(C^{\prime}\) with angle \(\gamma\). (c)(i) Add up the areas of these six lunes (two with angle \(\alpha,\) two with angle \(\beta,\) and two with angle \(\gamma\) ). Check that this total includes every part of the sphere at least once. (ii) Which parts of the sphere have been covered more than once? How many times have you covered the area of the original triangle \(A B C ?\) And how many times have you covered the area of its sister triangle \(A^{\prime} B^{\prime} C^{\prime} ?\) (iii) Hence find a formula for the area of the triangle \(A B C\) in terms of its angles \(-\alpha\) at \(A, \beta\) at \(B,\) and \(\gamma\) at \(C\)