CH. i] ARITHMETICAL RECREATIONS 7 



of V, and similarly find analogous numbers G, JD Rules 



for the calculation of A, B, G,... are given in the theory of 

 numbers, but in general, if the numbers a', b', c' ,... are small, 

 the corresponding numbers A, B, G, ... can be found by in- 

 spection. I proceed to show that n is equal to the remainder 

 when Aa + Bb+ Gc+ ... is divided by p. 



Let N = Aa + Bb + Gc + ..., and let M (x) stand for a 

 multiple of x. Now A = M (a') + 1, therefore Aa = Mia!) + a. 

 Hence, if the first term in N, that is Aa, is divided by a, the 

 remainder is a. Again, B is a multiple of a'c'd" .... Therefore 

 Bb is exactly divisible by a' Similarly Gc, Dd, . . . are each 

 exactly divisible by a'. Thus every term in N, except the first, 

 is exactly divisible by a'. Hence, if N is divided by a', the 

 remainder is a. Also if n is divided by a', the remainder is a. 



Therefore N-n = M(a). 



Similarly N-n = M(V), 



N-n = M(o'), 



But a', b', c',... are prime to one another. 



.-. N-n = M(a'b'c'...) = M(p), 

 that is, N=M (p) + n. 



Now n is less than p, hence if N is divided by p, the 

 remainder is n. 



The rule given by Bachet corresponds to the case of a' = 3, 

 V = 4, c'=5, _p=60, 4 = 40, 5 = 45, 0=36. If the number 

 chosen is less than 420, we may take a' =3, V = 4, c' = 5, d" = 7, 

 p = 420, A = 280, 5 = 105, G = 336, D = 120. 



To FIND THE RESULT OF A SERIES OF OPERATIONS PER- 

 FORMED ON ANY number (unknown to the operator) without 

 ASKING ANY questions. All rules for solving such problems 

 ultimately depend on so arranging the operations that the 

 number disappears from the final result. Four examples will 

 suffice. 



First Example*. Request some one to think of a number. 

 Suppose it to be n. Ask him (i) to multiply it by any number 

 you please, say, a ; (ii) then to add, say, b ; (iii) then to divide 

 * Bachet, problem Yin, p. 102. 



