FACTORS OF ANY GIVEN NUMBER. 
19 
let y be the difference between this middle number and each 
of these factors, so that the factors themselves are x + y and 
a; — y. Then the number itself which we will call a will be 
(^x + y) {x — y) ~ — tf. That is to say, it will be less 
than the square of the middle number by the square of the 
difference between the middle number and either of the 
factors. To illustrate this, let us take the number 10 and 
suppose it to be a middle number between certain pairs of 
factors equally distant from 10. 
Difference 
between 
100 and each 
product. 
Square root of 
ditto. 
Difference between 
J , 10 and each 
factor. 
10 
X 
10 
100 .. 
9 
X 
11 
99 .. 
1 
i 
*i 
8 
X 
12 
96 .. 
4 
2 
2 
7 
X 
13 
91 . . 
9 
3 
3 
6 
X 
14 
84 .. 
16 
4 
4 
5 
X 
15 
75 .. 
25 
5 
5 
and so on. 
It is evident at once that {x + xj) {x — ?/) or x^—y"^ is 
less than x^ for all values oi y. In other words, unless a, the 
given number, is a perfect square, in which case the factors 
are each of them yy/ff^ the middle number must be greater 
than \/ a, for the product of all numbers equally distant 
from j\/ a is of course less than a. Our first process therefore 
is to take the square root of a, and if this is not a perfect 
square, we take the next integral number above Va ; let this 
be b. 
"We then square b and deduct a. Then, if — a be a 
perfect square, we know at once that b is the middle number 
of the factors, and /\J b'^ — a is the difference between this 
middle number and each of the factors. Let us take an 
example : — 
Let the number be 45. The square root of this is 6 -|-, and 
the next higher number is 7. The square of this is 49, and 
