CONGRUENCE 113

1
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For example, (3 4 172) X 23 — (4 4+ 232) X 17
PR =1 %1

264. Th. Ifsa — tb = = 1, each of the integers s, a is prime
to each of the integers t, b.

[

265. Th. In the series of s's and t's given in § 254 each s,
except the last, is prime to its follower, each t, except the last, to
its follower, and each s to the corresponding {.

266. Th. Ifsa — tb = & (a R D), then s ]| ¢.
Or, In whatever way a (Q b is exprest in either of the forms
sa — tb or — sa + tb, s and t are prime to each other.

207. Th., ‘abFeA bl o g o
Since ab > ¢, c—= 0
Since b 1] ¢, two integers I and m can be found such that

 

1 =100+ mc
Then a = alb + amc
Since ab > ¢, alb > ¢
Also amc >> ¢
Therfor alb + ame >> ¢
Or G- ¢

268. Th. n>ha~n>hb~allb>Dn>> had
Since n >> ha, h—= 0 and n = gha

Then, since n >> hb, gha => hb
Hence ga > b
Hence, since a || b, g>b
Then gha > hab
Or n >> hab

269. Th. n>>a~n3bDn>$ (ab) : (aXRb)
We havea = kcand b = kd, wherea @ b = kand ¢ ]| d.
§ 236.

 

 

Since n >> aand n > b, n >> kc and n >> Ed.

Hence n >> kcd § 268.
Now kcd = (kc)d = a(b : k) = (ab) : k = (ad) : (a R b)
Therfor n > (ab) : (¢ R d)