This chapter deals with an arithmetic activity, trialled in the final year with students specialising in science. This teaching is supported by the solving of various problems: questions about prime numbers, cryptography, etc. The problem here is the factorisation of large numbers, which became vital for code breaking in the public interest (Delahaye, 2000). The idea comes from one of Fermat’s letters to Mersenne in which Fermat explains a method linked to a remarkable identity. This method was improved at the start of the twentieth century by Pierre and Eugène Carissan, they combined it with a sorting method using congruences. The Carissan brothers devised a congruence machine, completed in 1919, which mechanised the solution of certain Diophantine equations, in particular those associated with the problem of factorisation. The chapter gives elements of historical background on Fermat’s method and the Carissans’ machine explaining how the machine works and gives an account of the class activity with the problems the students were set and their reactions.