Chapter 6: Extra Divisibility & Primes
Perfect Squares, Cubes, etc.
1) Perfect squares have an odd number of total factors. 2) The prime factorization of a perfect square contains only even powers of primes. More Details: The numbers 4 (=2^2) and 25 (=5^2) are examples of perfect squares. One special property of perfect squares is that all perfect squares have an odd number of total factors. Similarly, any integer that has an odd number of total factors must be a perfect square. All other non-square integers have an even number of factors. Why is this the case? Think back to the factor pair exercises you have done so far. Factors come in pairs. If a and b are integers and a * b = c, then a and b are a factor pair of c. However, if c is a perfect square, then in one of its factor pairs, a equals b. That is, in this particular pair you have a * a = c, or a^2 = c. This "pair" does not consist of two different factors. Rather, you have a single unpaired factor: the square root. Notice that perfect squares are formed from the product of two copies of the same prime factors. For instance, 90^2 = (2 * 3^2 * 5)(2 * 3^2 * 5) = 2^2 * 3^4 * 5^2. Therefore, the prime factorization of a perfect square contains only even powers of primes. It is also true that any number whose prime factorization contains only even powers of primes must be a perfect square. Some examples: 144 = 2^4 * 3^2 40,000 = 2^6 * 5^4 By contrast, if a number's prime factorization contains any odd powers of primes, then the number is not a perfect square. For instance, 132,300 = 2^2 * 3^3 * 5^2 * 7^2 is not a perfect square, because the 3 is raised to an odd power. Perfect Cubes: The same logic used for perfect squares extends to perfect cubes and to other "perfect" powers. If a number is a perfect cube, then it is formed from three identical sets of primes, so all the powers of primes are multiples of 3 in the factorization of a perfect cube: for instance: 90^3 = 2^3 * 3^6 * 5^3
Primes (additional info)
1) There are an infinite number of prime numbers 2) There is no simple pattern in the prime numbers 3) Positive integers with exactly two factors must be prime, and positive integers with more than two factors are never prime Any integer greater than or equal to 2 has at least two factors: 1 and itself. Thus, if there are only two factors of x (with x equal to an integer greater than or equal to 2), then the factors of x must be 1 and x. Therefore, x must be prime. Also, do not forget that the number 1 is not prime. The number 1 has only one factor (itself), so it is defined as a non-prime number.
Factorials and Divisibility
Because N! is the product of all the integers from 1 to N, N! must be divisible by all integers from 1 to N. Another way of saying this is that N! is a multiple of all the integers from 1 to N. This fact works in concert with other properties of divisibility and multiples. For instance, the quantity 10! + 7 must be a multiple of 7, because both 10! and 7 are multiples of 7. Therefore, 10! + 15 must be a multiple of 15, because 10! is divisible by 5 and 3, and 15 is divisible by 5 and 3. Thus, both numbers are divisible by 15, and the sum is divisible by 15. Finally, 10! + 11! is a multiple of any integer from 1 to 10, because every integer between 1 and 10 inclusive is a factor of both 10! and 11!, separately.
Advanced GCF and LCM Techniques
Finding GCF and LCM using Prime columns Steps: 1) Calculate the prime factors of each integer 2) Create a column for each prime factor found within any of the integers 3) Create a row for each integer 4) In each cell of the table, place the prime factor raised to a power. This power counts how many copies of the column's prime factor appear in the prime box of the row's integer To calculate the GCF, take the lowest count of each prime factor found across all the integers. This counts the shared prime. To calculate the LCM, take the highest count of each prime factor found across all the integers. This counts all the primes less the shared primes. Try this problem: Find the GCF and LCM of 100, 140, and 250 Step 1: Calculate the prime factors of each integer. 100 = 2 * 2 * 5 * 5 = 2^2 * 5^2 140 = 2 * 2 * 5 * 7 = 2^2 * 5 * 7 250 = 2 * 5 * 5 * 5 = 2 * 5^3 Step 2: Create a column for each prime factor base and a row for each integer. The different prime factors are 2, 5, and 7, so you need three columns. There are three integers (100, 140, and 250), so you also need three rows. Step 3: Fill in the table with each prime factor raised to the appropriate power: _________________________________________________________ Number | 2 | | 5 | | 7 | 100 | 2^2 |*| 5^2 |*| 7^0 | 140 | 2^2 |*| 5^1 |*| 7^1 | 250 | 2^1 |*| 5^3 |*| 7^0 | To calculate the GCF, take the smallest count (the lowest power) in any column, because the GCF is formed only out of the shared primes. The smallest count of the factor 2 is one, in 250 (=2^1 * 5^3). The smallest count of the factor is 5 is one, in 140 (=2^1 * 5^1 * 7^1). The smallest count of the factor 7 is zero, since 7 does not appear in 100 or in 250. GCF = 2^1 * 5^1 * 7^0 = 10 To calculate the LCM, take the largest count (the highest power) in any column, because the LCM is formed out of all the primes less the shared primes. The largest count of the factor 2 is two, in 140 and 100. The largest count of the factor 5 is three, in 250. The largest count of the factor 7 is one, in 140. LCM = 2^2 * 5^3 * 7^1 = 3,500
Divisibility and Addition/Subtraction
General Rule: (Assume that N is an integer): 1. If you add a multiple of N to a non-multiple of N, the result is a non-multiple of N. (This same holds true for subtraction) For example: 18 - 10 = 8 (Multiple of 3) - (Non-multiple of 3) = (Non-multiple of 3) 2. If you add two non-multiples of N, the result could be either a multiple of N or a non-multiple of N. For example: 19 + 13 = 32 (Non-M of 3) + (Non-M of 3) = (Non-M of 3) 19 + 14 = 33 (Non-M of 3) + (Non-M of 3) = (Multiple of 3) The exception to this rule is when N = 2. Two odds always sum to an even DS example: Is N divisible by 7? 1) N = x - y, where x and y are integers 2) x is divisible by 7, and y is not divisible by 7 1) indicates that N is the difference between two integers (x and y), but it does not tell you anything about whether x or y is divisible by 7. Insufficient 2) tells you nothing about N. This statement is insufficient 1) & 2) combined indicate that x is a multiple of 7, but y is not a multiple of 7. The difference between x and y can never be divisible by 7 if x is divisible by 7 but y is not. Ans is C
Primes
Memorize all primes up to 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
GCF and LCM using Venn Diagrams Example
Q: Compute the GCF and LCM of 12 and 40 using the Venn diagram approach (see screenshot): The prime factorizations of 12 and 40 are 2 * 2 * 3 and 2 * 2 * 2 * 5 respectively. The only common factors of 12 and 40 are two 2's. Therefore, place two 2's in the shared area of the Venn diagram and remove them from both prime factorizations. Then, place the remaining factors in the zones belonging exclusively to 12 and 40. These two outer regions must have no primes in common. The GCF of 12 and 40 is therefore 2 * 2 = 4, the product of the primes in the shared area. The LCM is 3 * 2 * 2 * 2 * 5 = 120, the product of all the primes in the diagram. Note that if two numbers have no primes in common, then their GCF is 1 and their LCM is their product. For example, 35 (=3*7) and 6 (=2*3) have no prime numbers in common. Therefore, their GCF is 1 (the common factors of all positive integers) and their LCM is 35*6=210. Be careful: even though you have no primes in the common area, the GCF is not 0 but 1.
Prime Question (Understanding the meaning...):
Statement: There is no integer n such that x is divisible by n and 1 < n < x You are told that x is not divisible by any integer greater than 1 and less than x. Therefore, x can only have 1 and x as factors. In other words, x is prime. My own words: Take 17 for example: it's factors are 1 and 17 17 / \ 1 17 1 x There are no numbers (or n) in between 1 and 17 that is a factor of 17. 17 is not divisible by any numbers in between 1 and 17. This is true for all prime numbers.
Greatest Common Factor and Least Common Multiple
The Greatest Common Factor (GCF) is the largest divisor of two or more integers; this factor will be smaller than or equal to the starting integers. The Least Common Multiple (LCM) is the smallest multiple of two or more integers; this multiple will be larger than or equal to the starting integers. Finding GCF and LCM Using Venn Diagrams: For example, find GCF and LCM for 30 and 24: 1. Factor the numbers into primes: 30 = 2 * 3 * 5 24 = 2 * 2 * 2 * 3 2. Create a Venn Diagram 3. Place each shared factor into the shared area of the diagram (the overlapping area). In this case, it's one 2 and one 3 4. Place the remaining (non-shared) factors into the non-shared areas 5. The GCF is the product of the primes in the shared region: 2 * 3 = 6 6. The LCM is the product of all primes in the diagrams: 5* 2 * 3 * 2 * 2 = 120
Counting Total Factors
When a problem has a large number of factors, the factor pair method can be too slow. Q: How many different factors does 2,000 have? It would take a very long time to list all the factors of 2,000. However, prime factorization can shorten the process considerably. First, factor 2,000 into primes: 2^4 * 5^3. The key to this method is to consider each distinct prime factor separately. Consider the prime factor 2 first. Because the prime factorization of 2,000 contains four 2's, there are FIVE possibilities for the number of 2's in any factor of 2,000: none, one, two, three, or four. (Do not forget the possibility of no occurrences! For example, 5 is a factor of 2,000 and 5 does not have any 2's in its prime box.) Next, consider the prime factor 5. There are three 5's, so there are FOUR possibilities for the number of 5's in a factor of 2,000: none, one, two, or three. In general, if a prime factor appears to the Nth power, then there are (N + 1) possibilities for the occurrences of that prime factor. This is true for each of the individual prime factors of any number. You can borrow a principle from combinatorics to simplify the calculation of the number of prime factors in 2,000: when you are making a number of separate decisions, then multiply the number of ways to make each individual decision to find the number of ways to make all the decisions. Because there are five possible decisions for the 2 factor and four possible decisions for the 5 factor, there are 5 * 4 = 20 different factors. If you're working with more than two prime factors: If a number has prime factorization a^x * b^y * c^z (where a, b, and c are all prime), then the number has (x + 1)(y + 1)(z + 1) different factors. For instance, 9,450 = 2^1 * 3^3 * 5^2 * 7^1, so 9,450 has (1 + 1)(3 + 1)(2 + 1)(1 + 1) = 48 different factors
