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34 THE FASCINATION OF NUMBERS

4 3 2
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ference between two consecutive fifth powers is therefore of
the form 5(2x) +1=10x-+1, and the resulting number must
therefore always have 1 as its end-digit.

A table of end-digits for numbers up to the seventh powers
is given below. It will be seen that all fifth powers have the
same end-digit as the original number; all sixth powers have
the same end-digits as squares and all seventh powers have the
same end-digits as cubes. This process is continuous, every
power number having the same end-digit as the number four
degrees lower.

' as being of the form 2x where x=

End-digit of End digits of Power Numbers
Number Squares Cubes Fourth  Fifth Sixth  Seventh

—
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We have already given numerous relationships among
squares and cubes. The particular behaviour of numbers
consisting entirely of nines often points the way to discover-
ing different relationships.

Consider first, the following squares:

92=381
99*=9801
999*=998001
9999*=99980001

Any of the above square numbers can be obtained merely
by placing a g in front of the previous square and adding a
further nought between the digits 8 and 1. It will also be
noticed that if the square number is divided at the centre
into two separate numbers and these are added together,