2. Breaking down numbers into

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Presentation transcript:

2. Breaking down numbers into 1. Prime numbers 2. Breaking down numbers into Prime factors

Prime numbers are the basic building blocks of all number Useful when working with big numbers, such as in cryptography (secret codes) factoring very large number is hard and take computers a long time to do

Factors "Factors" are the numbers you multiply together to get another number: 2 X 3 = 6

Breaking down numbers into prime factors: Prime factorization "Prime Factorization" is finding which prime numbers multiply together to make the original number.

Example 1: What are the prime factors of 12 ? It is best to start working from the smallest prime number, which is 2, so let's check: 12 ÷ 2 = 6 Yes, it divided evenly by 2. We have taken the first step! But 6 is not a prime number, so we need to go further. Let's try 2 again: 6 ÷ 2 = 3 Yes, that worked also. And 3 is a prime number, so we have the answer: 12 = 2 × 2 × 3 As you can see, every factor is a prime number, so the answer must be right.

Example 2: What is the prime factorization of 147 ? Can we divide 147 evenly by 2? 147 ÷ 2 = 73½ No it can't. The answer should be a whole number, and 73½ is not. Let's try the next prime number, 3: 147 ÷ 3 = 49 That worked, now we try factoring 49, and find that 7 is the smallest prime number that works: 49 ÷ 7 = 7 And that is as far as we need to go, because all the factors are prime numbers. 147 = 3 × 7 × 7

Factor Tree "Factor Tree" can help: find any factors of the number, then the factors of those numbers, etc., until we can't factor any more.

48= 8 × 6, so we write down "8" and "6" below 48 Now we continue and factor 8 into 4 × 2 Then 4 into 2 × 2 And lastly 6 into 3 × 2 We can't factor any more, so we have found the prime factors. Which reveals that 48 = 2 × 2 × 2 × 2 × 3 (or 48 = 24 × 3 using exponents)

Unique: There is only one (unique!) set of prime factors for any number. Example The prime factors of 330 are 2, 3, 5 and 11: 330 = 2 × 3 × 5 × 11 There is no other possible set of prime numbers that can be multiplied to make 330. .