306 H. F, TALBOT ON A GENERAL SOLUTION OF 
becomes simply 4a —3 =12—3=9,or0. If we go on, we find with z= 4 
the reduct is 0; with z = 5 the same; but with z = 6 the reduct is 3. With 
“= 7 itis 0, and with z = 8itis0. That shows that if x is of the form 9a + n, 
and 2 has any value (except 6 or 0) the polynomial will be divisible by 9 (its 
reduct being 0). 
To show the great utility of these various theories, I will take the following 
example. In Barlow’s Mathematical Tables the number 380204032 is given as 
being a perfect fifth power. 
Let us therefore calculate its root. It may be said there is no rule given in 
books of arithmetic for the finding of fifth roots, and therefore we must have 
recourse to some tentative method. But that is not necessary; the following 
simple considerations lead directly to the desired result: 
1. Fifth powers end in the same digits as their roots. 
2. One-figure numbers have 1 to 5 figures in their fifth powers. 
Two-figure numbers have 6 to 10. 
Three-figure numbers have 11 to 15. 
Four-figure numbers have 16 to 20. 
3. The little table, which I have given in a former page, shows that if the 
reduct of a number is 1, the reduct of its fifth power is 1, the other digits giving 
the following results: 
Reduct of z 1 3 4 5 6 
Reduct of z® | 1 5 0 7 y 0 
bo 
= ~T 
8 0 
8 0 
Application to the given example. The given fifth power is 380204032, 
whese reduct is 4. 
1. This being a number of 9 figures, its root will have 2 figures. 
2. Since the fifth power ends in 2, the root ends in 2 likewise. Hence the 
root is one of the following series of numbers: 
12.22.32.42.52.62.72. 82.92. 
3. Since the reduct is 4, the reduct of its root is 7; in other words, the 
root is of the form 9x + 7. Hence the root is one of the following series of 
numbers: 
16.25.34. 45: of ole 70% 70s OS eur. 
But the only number common to the two series is 52. Hence 52 is the root 
required, 
It is rather curious to observe that it is much easier to find the fifth root of 
the given number than it would be to construct the fifth power if the root were 
given. This is opposed to the ordinary opinion that inverse processes are more 
difficult than direct ones. 
Another Example.—Find the fifth root of 1, 350, 125, 107, a number whose 

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