Mathematics. — “A problem of combinatorial analysis connected 
with the determination of the number of different ways in 
which the greatest common divisor of two products can be 
found’. By Prof. Freo. ScHen. (Communicated by Prof. D. J. 
KorTEWEG). 
(Communicated in the meeting of June 28, 1919). 
1. The greatest common divisor (G. C.D.) of the two numbers 
OS EON AD POD a ep es ade Ap) 
is obtained as the product of the mn greatest common divisors Gi; 
Mima ean (ande SL Zire os) LU 1S 10. be 
understood, that two numbers, whose G. C.D. is determined, are to 
be divided by that G.C.D. and that for the remaining part of the 
calculation the quotients are substitued for the original numbers. If 
such a quotient is combined with another number, we divide again 
py their G..C..D.-ete.’). 
So the mn numbers G;; are calculated in a definite order. Gi; 
ws the G. C. D. of the numbers a; and. b';, a’; being obtained by 
division of a; by all previously determined greatest common divisors 
that are connected with a;, while 6'; is derived in the same way 
from bj. 
a 
2. We now inquire into the nwnber of ways in which the G. C.D. 
of the numbers (1) can be calculated in the manner indicated 
m N°. 1. A different order of the mn numbers (;; does not neces- 
sarily yield a different manner of calculation of the number G 
sought, for two manners of calculation must of course be considered 
as identical, if they consist of the same calculations and only differ 
in the order in which the separate calculations are carried out. 
Hence it is immaterial which of the two numbers Gj and Grp is 
calculated first, provided «A and at the same time 7 #/. Conse- 
quently it is only the order of the numbers G;; which agree either 
in the first indices or in the second, that is essential. 
9: lhe special values- of the numbers @,,...,@mn, 0,,.:-., 6, may 
1) The last time that one of the numbers dj, Wg,.-., Ay, 01, Da, +.» bis com- 
bined with another the division of the first-mentioned number by the G. C. D. 
found may of course be dispensed with, the quotient being no further involved in 
the calculation. 
