The integers n and q are called factors of m . If the division does not come out even, the remainder is less than the number we tried to divide. For example, 16 divided by 5 is 3 with a remainder of 1. This whole collection of elementary school flashbacks can be summarized in a statement that sounds far more impressive than “long division,” namely, the Division Algorithm . The Division Algorithm. Suppose n and m are natural numbers. Then there exist unique numbers q (for quotient) and r (for remainder), that are either natural numbers or zero, such that m 5 nq 1 r and 0 r n 2 1 (r is greater than or equal to 0 but less than or equal to n 2 1). Prime Time Factoring a big number into smaller ones gives us some insights into the larger number. This method of breaking up natural numbers into their basic compo- nents leads us to the important notion of prime numbers. There certainly are natural numbers that cannot be factored as the product of two smaller natural numbers. For example, 7 cannot be factored into two smaller natural numbers, nor can these: 2, 3, 5, 11, 13, 17, which, including 7, are the first seven such numbers (let’s ignore the number 1). These unfactorable numbers are
Number Contemplation 70 called prime numbers . A prime number is a natural number greater than 1 that cannot be expressed as the product of two smaller natural numbers. The prime numbers form the multiplicative building blocks for all natural numbers greater than 1. That is, every natural number greater than 1 is either prime or it can be expressed as a product of prime numbers. The Prime Factorization of Natural Numbers. Every natural number greater than 1 is either a prime number or it can be expressed as a product of prime numbers. Let’s first look at a specific example and then see why the Prime Factorization of Natural Numbers is true for all natural numbers greater than 1. Is 1386 prime? No, 1386 can be factored as 1386 2 693 H11005 H11003 . Is 693 prime? No. It can be factored as 693 3 231 H11005 H11003 . The number 3 is prime, but 231 3 77 H11005 H11003 . So far we have 1386 2 3 3 77 H11005 H11003 H11003 H11003 . Is 77 prime? No, 77 7 11 H11005 H11003 but 7 and 11 are both primes. Thus we have 1386 2 3 3 7 11 H11005 H11003 H11003 H11003 H11003 . So, we have factored 1386 into a product of primes. This simple “divide and conquer” technique works for any number! Proof of the Prime Factorization of Natural Numbers Let n be any natural number greater than 1 that is not a prime. Then there must be a factorization of n into two smaller natural numbers both greater than 1 (why?), say n 5 a b . We now look at a and b . If both are primes, we end our proof, since n is equal to a product of two primes. If either a or b is not prime, then it can be factored into two smaller natural numbers, each greater than 1. If we continue in Understanding a specific case is often a major step toward discovering a general principle.
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