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all that is thus required is for the right summand in parentheses, 8*1 + 1*1 + 0*1
to be divisible by 3. Hence, the rule of three, for determining whether or not
a given integer is divisible by 3 becomes clear.
A corresponding rule of nine is not difficult to understand, as are numerous
other shortcuts for determining divisibility criteria. Clearly, one needs to test
candidate divisors only up to the square root of the number in question.
However, when one is faced with an image on the screen, it would be nice
not to have taken the trouble (however little) of finding that 810 is divisible
by 3, only to find that 922 is not divisible by 3. A far better approach is to
rewrite each number using some systematic procedure and then compare a
pair of expressions to determine, all at once, which numbers are divisors of
both 810 and 922. For this purpose, the Fundamental Theorem of Arithmetic
is critical.
Fundamental Theorem of Arithmetic (e.g., Gauss, 1801; Herstein, 1964): Any
positive integer can be expressed uniquely as a product of powers of prime
numbers (numbers with no integral divisors other than themselves and 1).
Thus, in the 810 by 922 example, 810 = 2*405 = 2*5*81 = 2*3*3*3*3*5 (super-
scripts avoided by repetitive multiplication) and, 922 = 2*461. The number
810 was easy to reduce to its unique factorization into powers of primes; one
might not know whether or not 461 is a prime number or whether further
reduction is required to achieve the prime power factorization of 922.
To this end, the Sieve of Eratosthenes (the same Librarian at Alexandria who
measured the circumference of the Earth, described in Chapter 2) works
well. To use the sieve, simply test the prime numbers less than the square root
of the number in question. The square root of 461 is about 21.47. So, the only
primes that can possibly be factors are: 2, 3, 5, 7, 11, 13, 17, and 19. Clearly, 2,
3, and 5 are not factors of 461. A minute or two with a calculator shows that
7, 11, 13, 17, and 19 are also not factors of 461.
Notice that these calculations need only be made once—when one has the
unique factorization, all divisors of both numbers are known from looking at
the two factorizations, together. Thus:
810 = 2*3*3*3*3*5; 922 = 2*461
and 2 is the only factor common to both numbers, so that 1/2 is the only scal-
ing factor for this image.
4.7 Preservation of the aspect ratio
Because the only factor the two numbers 810 and 922 have in common is 2,
it follows that the only scaling factor that can be used, that will preserve the
aspect ratio, is 1/2. Had 1/3 or some other ratio been employed, a distorted
view of the original image would have been the result.
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