( 330; ) 
Le. equal to 4911. On the other hand the factor 953 gives 
4772 — 476%, ie. 953 X 1 and appears to be a prime number. 
The last operation is rather long, a number of 446 numbers 
having to be added; as for larger numbers the number of additions 
nearly increases proportionally it is peremptorily necessary to seek 
for a shorter method. , 
Such an abbreviation can be derived from the consideration of 
last two figures of the number to be factorised. 
For a square must terminate in one of the following pairs of 
figures: 00; O1, 21, 41, 61, 81; 04, 24, 44, 64, 84; 25; 16, 36, 
46,576,; 96:00, 29, 49.69 8D. 
The last pair of figures of the number N being 53 we have to 
determine out of the given pairs those pairs which, by addition 
of 53, furnish another pair. 
So one sees immediately that 6? only can terminate in one of 
the pairs 16, 36, 56, 76, 96, in which cases a? terminates in 
69, 89, 09, 29, 49 respectively. 
So we can shorten our algorithm to: 
ete. 
Now, as is immediately evident 67 + 69 + 71 + 73=4 70, 
and 75 + 77 419 +81 483 + 85 = 6 X 80; as similar groups 
present themselves over and over, we can shorten still more as 
follows: 7 
