DN O R RS T
Fa A Y 402 aaiinlad

 

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52 THE FASCINATION OF NUMBERS
so that Map+1=10(X,_,)+1
and Man=10(X,_,)

from which we show that X,,_, is a multiple of z, unless 7 is
a factor of 10.

Every prime number greater than g differs from a multiple
of both 4 and 6 by a difference of 1 (either added or sub-
tracted). Unfortunately it does not follow that every number
so related to 4 and 6 is necessarily a prime.

The prime

19=(3 X6) +1=(5x4) -1
whereas 55=(9 x6) +1=(14X4)—1

and the number 55 is not a prime.

In an endeavour to find a formula for the generating of
primes, Euler proposed the form n?-+n-41. This gives a
prime number for all values of z up to 39 but obviously fails
for n=41. The expression n%—n-41 gives similar results.

It is worthy of note that no triangular number can be a

(n) (n+1)

prime since each such number, being of the form ~——-~,
2

clearly has the factors (z) and ——(n—:l).

The usual way of finding the highest common factor of
any two numbers is to reduce the numbers to their con-
stituent factors and then to select those factors which are
common to both. A more interesting way, discovered by
Euclid, is based on the proposition that if two numbers have
a common factor then the difference between them also has
the same factor. The process is therefore one of expressing
one number in terms of the other and then treating the
second number and the remainder in a similar way.

In order to find the highest common factor of 6600 and
2424, the number 6600 is first expressed as a multiple of 2424
plus a remainder R,. The number 2424 is then expressed as a
multiple of R, plus a new remainder R,. The number R, is
then expressed as a multiple of R, plus a third remainder R,.

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