Each question is worth 7 marks
Time allowed is 4 hours
No books, notes or calculators permitted
Give full proofs with your answers
1) Prime numbers , and satisfy the following two conditions.
Find the largest possible value of .
2) Find all solutions in non-negative integers to the equation
3) An example of Clayton's cancelling is . That is, the correct result is obtained using the incorrect method of "cancelling" the 6s. Find all instances of Clayton's cancelling which simultaneously satisfy the following criteria.
(i) Both numerator and denominator are strictly two-digit numbers with the numerator smaller than the denominator.
(ii) The units digit of the numerator is equal to the tens digit of the denominator.
(iii) Crossing out the units digit of the numerator and the tens digit of the denominator yields the correct lowest terms simplification of the original fraction.
4) Find all solutions in positive integers , and to the equation
5) Which positive integers can be written in the form
for positive integers ?