You can prove that multiplying any odd number by 3 always gives you an odd number if
you realize that the last digit of an odd number is always odd. Think of an odd number.
However large it is, it looks like one of the following:
__1
__3
__5
__7
__9
where the long underscore represents any string of digits you want to put there. Now
think of “long multiplication” by 3. Remember how you arrange the numerals on the
paper and then do the calculations. You always start out by multiplying the last digits of
the two numbers together, getting the last digit of the product. The odd number on top,
which you are multiplying by the number on the bottom, must end in 1, 3, 5, 7, or 9.
If the number on the bottom is 3, then the last digit in the product must be 3, 9, 5 (the
second digit in 15), 1 (the second digit in 21), or 7 (the second digit in 27) respectively.
Therefore, any odd number times 3 is always odd.
- To figure out whether or not 901 is prime, try to break it down into a product of prime
factors. If you succeed in doing that, then 901 is not prime. If you fail, then 901 is
prime. First take its square root using a calculator. You’ll get 30 with a decimal point
and some digits. Round this up to the next whole number, which is 31. Using
Table 3-1, you can list the set all the primes up to and including 31. That set is
{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31}
Now, divide 901 by each of these numbers. As things turn out, 901 is divisible by 17
without a remainder. Therefore, 901 is not prime.
- To find the prime factors of 1,081, start by taking its square root using a calculator.
You should get 32 with a decimal point and some digits. Round this up to 33. Using
Table 3-1, list the set all the primes less than or equal to 33. That set is
{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31}
Divide 1,081 by each of these numbers. You will see that 1,081 is divisible by 23 without
a remainder. You get
1,081= 23 × 47
Both of these factors are prime, as you can see by looking at Table 3-1. The number 1,081
therefore has prime factors of 23 and 47.
Chapter 3 593
