NUMERICAL EQUATIONS OF ALL DEGREES HAVING INTEGER ROOTS. 309 
6 for w in the original equation and the result is nine times 11200, the reduct of 
which number is 4, therefore Q = 4. Solve the equation Py = Q or 8y = 4. 
To do this we must add successive nines to 4 till it becomes divisible by 8. 
Thus, 4, 13, 22, 31, 40, which last number is divisible. Hence Py = Q means 
8y = 40, whence y = 5. Finally 
a=9y+n=97+6=9.54+6=51. 
Therefore the second root of the equation is 51. 
Let us now proceed to find the third root. Let x = 7, then the formula 
p = 3n* + 8 becomes p = 3.49 + 8 = 155, whose reduct is 2, therefore P = 2. 
Substitute 7 for x in the original equation and the result is nine times 10,472, 
whose reduct is 5, therefore Q@=5. Solve the equation Py = Q or 2y = 5, which, 
by adding 9, is 2y = 14, and we obtain y = 7. Finally,w = 9y +n=9y+7= 
9.7 +7=70. Therefore the third root of the equation is 70. 
Thus we have determined the three roots of this cubic to be 41, 51, and 70. 
It is now easy to verify these roots by multiplying together the three factors 
a — 41, « — 51, and « — 70, thus— 

z— 41 
x«— 51 
vt — Al.z 
— 5l.a% + 41.51 
e2— 92 w + 2091 
za — 70 
xz? — 92 2% + 2091 x 
— 70 #2 + 70.92 « — 70.2091 
x® — 162 2% + 8531 aw — 146370 

which was the given equation. 
Second Example. 
Let the given equation be 
av —174 2? + 9749 x — 177276 = 0. 
The reduct equation will be 
uv — 3a + 2a—3=0. 
Hence the coefficients a=3, 6=2. Proceeding as before to try the nine digits. 
1 does not satisfy this equation, for it gives 1 — 3 + 2 — 3, which is not equal 
to 0. Neither does 2 satisfy it, for it gives 8 — 12 + 4 — 8, which is not equal 0. 
Trying the others, we find that the equation is only satisfied by the numbers 
6, 7, and 8. We verify this result by multiplying together (« —6) (# — 7) 
(z — 8) = 0, which gives the reduct equation «* — 3x7 + 2a —3 = 0 as before. 
Let us now calculate the three roots of the cubic, which we have found to be of 
the forms 9y + 6, 9y/ + 7, 9y” + 8. 
First try the form « = 97 + 6, so thatn =6. Put 2 =6 in the original 
equation and the result is nine times 13870, whose reduct is 1 = Q. 
