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If you see a large natural number, you can sometimes tell right away that it s not prime If its numeral ends in an even number, you know that one of its factors is 2, so it can be factored into a pair of natural numbers other than 1 and itself Sometimes, numbers seem at first as if they ought to be prime, but it turns out that they are not A good example is 39 It can be factored into 13 3 Another is 51, which can be factored into 17 3 Still another is 57, which can be factored into 19 3
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Table 3-2 The first 24 perfect squares The numbers 0 and 1 are included here When you take the square root of a perfect square, you always get a natural number Note that we start with the 0th rather than the 1st in order here That way, the order agrees with the number squared
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Order 0th 1st 2nd 3rd 4th 5th 6th 7th Square 0 1 4 9 16 25 36 49 Order 8th 9th 10th 11th 12th 13th 14th 15th Square 64 81 100 121 144 169 196 225 Order 16th 17th 18th 19th 20th 21st 22nd 23rd Square 256 289 324 361 400 441 484 529
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42 Natural Numbers and Integers
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The best way to find out whether or not a large odd number is prime is to try to factor it into primes If the only factors you get are itself and 1 (ie, if you can t factor it into primes), then your number is prime There are some other techniques you can use determine when a number is not prime, such as the divisibility tricks you ll see later in this chapter
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When an even number is multiplied by 7, the result always even Show why this is true
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For the first few even natural numbers, multiplication by 7 always gives you an even number Here are the examples for all the single-digit even numbers: 0 7=0 2 7 = 14 4 7 = 28 6 7 = 42 8 7 = 56 You can prove that multiplying any even number by 7 always gives you an even number if you realize that the last digit of an even number is always even Think of an even number p any even number This number p, however large it might be, must look like one of the following: ______0 ______2 ______4 ______6 ______8 where the long underscore represents any string of digits you want to put there Now think of long multiplication by 7 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 even number on top, which you are multiplying by the number on the bottom, must end in 0, 2, 4, 6, or 8 If the number on the bottom is 7, then the last digit in the product must be 0, 4 (the second digit in 14), 8 (the second digit in 28), 2 (the second digit in 42), or 6 (the second digit in 56) respectively The product of any even number and 7 is therefore always even
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Natural Number Nontrivia
Here are some interesting facts about natural numbers I was about to call them trivia, but after thinking about it for awhile, I decided that the ones involving primes are not trivial at all!
Natural Number Nontrivia
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 calculator 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 division, 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 number 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
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