Worcester daily spy. [volume] (Worcester [Mass.]) 1850-1888, 1858-03-11


MesSSS Editors In last weeks Spy is re-
quest for explation Of the fact that the differ
ence between two numbers consisting of the
same figures differently arranged, is always null
tiple of 8. The explanation requires the demon
stration Of two other propositions, as follows
Prop LIST. If to ang figure of any number any
one Of the nine digits be added, and the same dig
1t be taken from any other figure of that number
number will be produced, the difference between
which and the original number will always be o
a multiple of9.
Thus let the original number be 672
Take s from 2, and add S-to g. and we have
1904 and 1864 minus 672 equals 1872, which is just
% times y. Or, add to 2, and take from g.
and the result, l77, taken from 672, leaves 495,
which is just 90 times 9. To understand the
reason of this, it must be remembered that 10
consists OF and and that any number of tens
consists OF the same number Of Mines together
with the same number of units. Thus 90 which
Is tens, consists off mines and units 70 or
terIs contains mines and Z units and 1900 or 200
tens contains 200 mines and 200 units. This pro-
perty of numbers may be well pondered, and when
understood will be seen to establish Prop 2nd
VIZ: Ii from a number, expressed BY any one of
the digits aud any number of cphers annexed, we
subtract another number, expressed by the same
digit and smaller number of ciphers, the re-
mainder will always be 8 or a multiple of g.
Thus 1000 minus 700, will be multiple of 8, be-
cause 7OV0 which is 200 tens consists of just 700
mines and 700 units; 5000 minus so will be mul.
tiple of 9; for 5000 or 800 tens equals 500 mines
plus 800 units, and 800 units or so tens equals so
mines plus 81 units therefore 5000 equals $50
mines pins 50 units In the same manner it can
be shown that the difference between Any other
two numbers expressed according to Prop End,
will be 8 or multiple of g. Now let 500 he treat.
ed according to Prop. lst, by taking from and
adding it to6 and it is manifest that the num-
Der added to 693 by the operation, is 700 minus
% which, by Prop. End, is a multiple of o; or if
5 be taken from g. and added to g. the number
subtracted from SOB will be 500 minus 5, a multi.
pie oi9 Again treat 5T2 by adding to 2 and
taking 1t from 7, and it will be seen that the
amount taken from 572 is go minus 6, multiple
of9.
Every case of transposed numbers may be class
ed under Prop. lst or under Prop ID and the
following self evident axiom, viz
II, OF three numbers, the difference between the
LIST and End is multiple of 8, and the difference
between the I'd aud 80 is a multiple of 9, then
the difference between the lst and Sd will be
multiple OF y Thus 70 transposed is 81, which is
made from 70 By adding to aud taking it from
z, or by taxing 40 minus from 70. 5S transpos-
ed is 80, which is made by adding 30 minus to
%: The transpositions Of 932 are 923, 392 329,
go and 239 923 is made by taking 10 minus l
from 992 1872 is made by taking 800 minus 60
from 63z 329, by taking 600 minus from 923
49, BY taking 700 minus 20 from 923 and 239,
BY taking 700 minus from 932 Permit in clos.
ing another proposition for the amusement of
the curious. If any of the digits be taken,
from, or added to, the right hand figure of any
number and this digit be divided into anv adj.
missable number of parts, and each part be added
TO- or taken from, one of the other figures of the
number, the difference between the number thus
produced and the original number will be Inul.
tiple of 3. Thus take from the unit figure of
5f9 and add and respectively to and Z, and
we have 99l and 99l minus $40 will Se null tiple
of9.