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%g

32 THE FASCINATION OF NUMBERS

This enables us to reduce the form of the common differ-
ence in the nth power series from (n)(z—1)(n—2) . .. (2)(1)
to the simple expression 7!

The next regular pattern to appear in ‘power’ numbers is
that revealed by their end-digits—that is, the last digit in
each number. If the squares of consecutive integers are
written out as follows, there are two points which are imme-

diately apparent.
100 400

1. o8 -asr. g6 . 441
4 64 144 324 484
9 49 169 289 529
16 36 196 256 576
25 225 625

These numbers have been placed consecutively in columns
which show the numbers downward in the odd columns and
upward in the even columns. This makes it easier to check
that in each set of ten numbers the end-digits are, in order,
1, 4,9, 6, 5,6, 9, 4, 1, 0, and that, moreover, of this set of
numbers the digits preceding the digit 5 appear again after 5,
but in reverse order.

These end-digits repeat in the same order indefinitely as
each new square number is reached, so that it may always be
stated with confidence that no number whose end-digit is
2, 3, 7 or 8 can possibly be a square number.

It may be wondered why two numbers whose end-digits
are 9 and 1 respectively each give numbers with the end-
digit 1 when squared. There is an elementary reason why
this should be so and, stated simply, it may be said that it is
because both numbers differ from a multiple of 10 by 1, and
is based on the fact that ( —1)2 gives the same result as (4 1)2.

Any number ending in g can be expressed as a multiple of
10, less 1 (that is, 1on—1). Any number ending in 1 can be
expressed as a multiple of 10, plus 1 (that is 10n+1). If these
two expressions are squared, the results are:

(ton—1)2=100n%—2(10)n+1
(ron+1)2=100n2+2(10)R+1I