Divisibility
If you want to know whether or not a large number can be divided by a single-digit number
without leaving a remainder, there are some handy little tricks you can use. You can use a cal-
culator to see immediately whether or not any number is “cleanly” divisible by any other, but
the following rules can be interesting anyway.
- A natural number is divisible by 2 without a remainder if it is even.
- A natural number is divisible by 3 without a remainder if the sum of the digits in its
numeral is a natural-number multiple of 3. - A natural number is divisible by 5 without a remainder if its numeral ends in either
0 or 5. - A natural number is divisible by 9 without a remainder if the sum of the digits in its
numeral is a natural-number multiple of 9. - A natural number is divisible by 10 without a remainder if its numeral ends in 0.
You can combine these tricks and get the following facts:
- A natural number is divisible by 4 without a remainder you get an even number after
dividing it by 2. - A natural number is divisible by 6 without a remainder if it is even and the sum of its
digits is a natural-number multiple of 3. - A natural number is divisible by 8 without a remainder if you can divide it by 2 and get
an even number, and then divide that number by 2 again and get an even number.
There aren’t any convenient tricks, other than using a calculator or performing “long divi-
sion,” to find out if a natural number is divisible by 7 without leaving a remainder.
Is there a largest prime?
Now that you know what a prime number is, and you know that any nonprime natural num-
ber can be broken down into a product of primes, you might ask, “Is there a largest prime?”
The answer is “No.” Here’s why. You might have to read the following explanation two or
three times to completely understand it. Try to follow it step-by-step. If you can accept each
step of this argument one at a time, that’s good enough. The fact that there is no such thing
as a largest prime is one of the most important facts, or theorems, that have ever been proven
in mathematics.
Let’s start by imagining that there actually is a largest prime number. Then we’ll prove
that this assumption cannot be true by “painting ourselves into a corner” where we end up
with something ridiculous. Now that we have decided there is a largest prime, suppose we give
it a name. How about p? Theoretically, we can list the entire set of prime numbers (call it P).
It might take mountains of paper and centuries of time, but if there is a largest prime, we can
eventually write all of the primes. We can describe the set P in shorthand like this:
P= {2, 3, 5, 7, 11, 13, ..., p}Suppose that we multiply all of these primes together. We get a composite number, because
it is a product of primes. No doubt, this number is huge—larger than any calculator can
Natural Number Nontrivia 43