NUMBERS. 
Properties of Numbers. 
Prop. I. 
On the Divi ae of — prop number N, when 
reduced to the * di, &c. a, b, cy &e. 
being prime number, wil a the number rs the aivifors 
exprefled by the for 
(m+ 1) oe . (ptr) (g+1)> &e.3 
the uel itfelf being confidered as one of its divifors, 
uppofe, for example, it were ial to find how many 
divifors belong to number 360 
r ba ave ee es 53 therefore m= 3, n=2 
pote Hencey (443) (240) UE) = 40308 = ah 
the eae of j its divifors. 
Again: required how many divifors belong to the num- 
ber 1000. 
i 000 = 23.5% ak m= 3, and 2 = 33 
whence (3+1) (3 +1) =4-4=16, the en vo di- 
vifors required, whack are as Allow: I, 2) 4y 5» 8, 10, 205 
25> 40, 5° 100, 1295 200) 2505 5005 1ooo. 
Prop. LI. 
, fos find a number that fhall have any required number of 
vifors. 
w reprefent the given number of divifors: refolve ev 
= ts ators epenmen, oat &c. Then take m= x 
» p=2z—), ; fo thall a” 3" c &c. be 
ean nu ae t AE where a, by c, &c, may be taken any 
prime aac at pleafur 
Exam.—Find a panibe that fhall have 30 divifors. 
Oe e455: that is, w= 2, y eos 
therefore, m= 2, p= 4; whencea. é. cis anume 
ber having 30 divifors, as required. 
Firft,. 
If a= 2, b= 3, c= 53 then 2. 3°. 5*= 11250. 
pa b=3; c= 2; then 5.37. 2= 720. 
Each of which numbers has 30 divifors; and it is evident 
that various other numbers might be obtained, that would 
ne the fame property, by only changing the values of a, 
1 Cy ne 
ponent the Jeaft root, the next greater exponent the next 
lefs root, &c.; the roots themfelves being the leaft prime 
numbers that can be employed for fon gl and which 
will of courfe depend upon the num 
Suppofe, for gree it were anid a ae the leaft 
number having 370 divi 
Here the greateft ae of factors is balla we — 
oh aay or when x= 2, = 
ore m= 1, n= pee i tt a be is the 
lea pa 5 and by makin e hav 
» $= 240, which is the  Tealt cme “that has 20 ai. 
Prop. II. 
Ds find the fum af all the divifors of any given nume 
"Ref olve the given number into the form a”. b".c? &c. 
ya . fum of all its divifora will be exprefled by the 
ormula 
7b! oe = J Bt! o. I : the ee 
a~t b—j ¢~} “) : 
Suppofe, for example, it were required to find the fum 
of all the divifors of 360, the number itfelf being included 
as one of them. 
e 360 = 2). 37.93 therefore a= 2, b= 3, c= 53 
n= . n= 35 p=ts whence, 
9* — poems 2. Y 
ay : a x 3 = 15+13.6= 131703 
2-—I1 —I 
3-1 
which is the fum of all the divifors of 360, the number itfelf 
being confidered as one of them. 
Prop. IV. 
To find how. many ae there are lefs than a given 
number N, and alfo prim 
Refolve the given aunt ae the form N= a™ 6" ¢ 
&c. then will 
—-1 b-1 ees I 
b ¢ 
exprefs the number i integers that are lefs than », and alfo 
prime to 
Exam.—How _many — are there lefs than 100) 
which are alfo prime to to 
Firft, 100 = 2” = 
a 
N x 9 ke. 
therefore, 
100 x 2 x SD = 4, 
the number fought, thefe being as follow, wiz. 
O30 
snl 
Ke) 
Ww 
~ 
p> 
w 
wi 
~I 
nN 
O 
(os) 
La 
ie} 
wo 
1I 
Exa. — How meu ean are om lefs than 360, 
that are vali is to it 
aa 53 therefore, 
— 3—t x fh = 96, 
2 3 5 
the number fought. 
Pror. V. 
A number that is the fum of two fquares prime to each 
other; can only be divided by numbers that are alfo the fums 
of two {quares ; which is the fame, every divifor of a 
number falling under the form # +4’, t and w being prime 
to each other, is itfelf alfo of the fame form 
us, for example, 65 = 8° + 1°, can only be divided 
a - which is the fum of two {quares 5 
i aes =2'4 17, 
Alfo, so= 7’ - a hase for divifors, 
And the fame for all other numbers falling under the above 
form, ee only that the two {quares muft be prime to 
each other 
Prop. VI. 
Thus 
