CONGRUENCE 141

Thus, if the radix is ten and modulus 7> — 1, or 99,
27,36,47 =27+ 36 +47=1,10=1+4 10 = 11;
if the radix is ten and modulus #® — 1, or 999,
684,329 = 684 + 329 = 1,013 = 14.

For proof we have
1
a = po+ pir™ + part™ oo pple = F (D)
k=0

where for all values of & either p,—< 0 or p,—> 0 and, since
a has more than one m-figure period, / > 0 and p;—= 0.

Now, since m > 0 and 2 = 0, km = 0.
Therfor, since r > 1, rew =1
Sincel >0and m > 0,Im > 0
Therfor g >l
Again |pr] =0
Hence, since 7" = 1, | pr| 7" = | P
Also |pil > 0
Hence, since '™ > 1, | pi]rim > | pi
Thus we have, putting # =0, 1, 2, ---, ],
|po| = |po]
|pr|r™ = | Pl
|p2| 7™ = | ol
|prlrim > [l
1 1
Adding we get X (|px|7*™) > 2 | pxl
k=0 k=0

But la| = =I§)(lpklf’k"‘)

 

 

l
kgo (par®™)

 

l l
and ’Zfik = 3 | Pl
k=0 k=0