INDETERMINATE PROBLEMS, 
197 
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'b in 
true arithmetical progression ; whereof the common dif 
ference being 18, and the number of terms = — — 
= 48, the sum will therefore be given = 20880 : to 
which adding 13485, the number of answers when q 
was less than 62, the aggregate 34365 will be the whole 
number of all the answers required. 
PROBLEM XIV, 
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To determine how many different ways it is possible to 
pay lOOOl. without using any other coin than croions, 
guineas, and moidores, 
By the conditions of the problem we have 5x 4- 21 y 
4- 27 z — 20000; where taking z = o, x is found 
= 4000 — 4y — and from thence the least value 
of y = O (0 being to be included, here, by the question); 
whence the greatest value of x is given =4000. More 
over, from the equation 5m 4- 21 n — 27, we have 
m = 5 — 4n — n ~.—\ from which n = 2, and m = 
5 
— 3: so that the general values of x and y, given in 
the preceding problem, will here become 4000— 2lq 
4- 3s, and 5q — 2z. Moreover, from the given equa 
tion, the greatest limit of 2 appears to be = " Q0Q - rz 
, , g — mz 4000 4-3x 740 
740; whence we also have 5—= = — — 
b 21 
= 296 = the greatest limit of <7; and = 1222 — 
190, expressing the lesser limit of <7, when the value of 
t, answering to some interpretations of 2, will become 
negative, while those of у will continue ailirmative. 
To find the number of all these affirmative values, up 
to the greatest limit of <7, let 0, 1,2, 3, 4, 5, &c. be 
now wrote in the room of <7 (as in the margin). Whence 
it is evident that the said number is composed of the 
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