‘PRIMES AND FACTORS’ 53

This process is continued until the first of any two consecu-
tive remainders is an exact multiple of the second.

6600 =2 x 2424 +1752
2424 =1 X 1752+ 672
1752=2 X 672+ 408
672=1 X 408+ 264
408 =1 X 264+ 144
264=1 X 144+ 120
I144=1X I20+ 24
I20=HX 24+ O

The remainder 120 is an exact multiple of the next remainder
24, and this last number is therefore the highest common
factor of the two original numbers.

Provided that a number can be resolved into its con-
stituent basic factors, there is a formula for calculating the
number of different factors of that number. The number is
first reduced to its factors in their lowest forms (a)* and (b)Y
where x and y are the indices representing the powers of each
factor included. For example, the number 24 =(22) (3?). The
formula then employed is based upon the values of the
indices of the factors; that is, on x# and y. Each index is in-
creased by 1 and they are then multiplied together to give
the required total number of factors. For the number 24,
where the indices are g and 1 respectively, the expression is
(3+1) (1+41)=8. These eight factors are 1, 2, 3, 4, 6, 8, 12
and 24.

This formula is of assistance in connexion with the check-
ing of amicable numbers (see Chapter g), and from it can be
derived another formula for calculating the product of all
the different factors of any number. If the number be z» and
the number of different factors be x, then the product of all
these different factors is #/2 or v/n®. Thus the product of all
the different factors of 24 equals 24* or 331,776.

The sum of all these different factors may also be found
rapidly. There are, however, two different methods depend-
ing upon the nature of the indices of the basic factors

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