6 ARITHMETICAL RECREATIONS [CH I 



remainder when a - b + 3 (c - d) is divided by 9, we have 

 x = e, y = 9 — r. 



The demonstration is not difficult. Suppose the selected num- 

 ber is 9a; + y. Step (i) gives 90a: + lOy + a. Let y + a = 3n + b, 

 then the quotient obtained in step (ii) is 30x + 2y + n. Step 

 (in) gives 300a; + 30y + 10n, + c. Let n + c = 3m + d, then the 

 quotient obtained in step (iv) is 100a; + lOy + 3n + m, which I 

 will denote by Q. Now the third digit in Q must be x, because, 

 since y if- 8 and a $■ 9, we have n ^ 5 ; and since n^-5 and c ^ 9, 

 we have m ^ 4 ; therefore lOy + 3n + m $■ 99. Hence the third 

 or hundreds digit in Q is x. 



Again, from the relations y + a = 3w + b and n + c = 3m + d, 

 we have 9m — y = a-b + 3(c — d): hence, if r is the remainder 

 when a — b + 3 (c — d) is divided by 9, we have y = 9 — r. [This 

 is always true, if we make r positive ; but if a — b + 3 (c — d) 

 is negative, it is simpler to take y as equal to its numerical 

 value ; or we may prevent the occurrence of this case by 

 assigning proper values to a and c.J Thus x and y are both 

 known, and therefore the number selected, namely 9x + y, is 

 known. 



Fifth Method*. Ask any one to select a number less 

 than 60. Eequest him to perform the following operations, 

 (i) To divide it by 3 and mention the remainder ; suppose it 

 to be a. (ii) To divide it by 4, and mention the remainder; 

 suppose it to be b. (iii) To divide it by 5, and mention the 

 remainder; suppose it to be c. Then the number selected is 

 the remainder obtained by dividing 40a + 456 + 36c by 60. 



This method can be generalized and then will apply to any 

 number chosen. Let a', 6', c', ... be a series of numbers prime 

 to one another, and let p be their product. Let n be any 

 number less than p, and let a, b, c, ... be the remainders 

 when n is divided by a, b', c', . . . respectively. Find a number 

 A which is a multiple of the product b'c'd' . . . and which 

 exceeds by unity a multiple of a'. Find a number B which is 

 a multiple of a'c'd' ... and which exceeds by unity a multiple 



* Bachet, problem vi, p. 84 : Bachet added, on p. 87, a note on the previous 

 history of the problem. 



