# Number Theory.  A prime number is a natural number greater than 1 that has exactly two factors (or divisors), itself and 1.  Prime numbers less than.

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Number Theory

 A prime number is a natural number greater than 1 that has exactly two factors (or divisors), itself and 1.  Prime numbers less than 50 {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47} * “2” is the only EVEN Number

Rules of Divisibility 285The number ends in 0 or 5. 5 844 since 44  4 The number formed by the last two digits of the number is divisible by 4. 4 846 since 8 + 4 + 6 = 18 The sum of the digits of the number is divisible by 3. 3 846The number is even.2 ExampleTestDivisible by

730The number ends in 0.10 846 since 8 + 4 + 6 = 18 The sum of the digits of the number is divisible by 9. 9 3848 since 848  8 The number formed by the last three digits of the number is divisible by 8. 8 846The number is divisible by both 2 and 3. 6 ExampleTestDivisible by

 Write the prime factorization of 663.  The final quotient 17, is a prime number, so we stop. The prime factorization of 663 is 3 13 17 13 3 17 221 663

 The greatest common divisor of two or more integers can be obtained in three steps:  STEP 1: Find the prime factorization of each integer. (Use Division Method)  375 = 3 × 5 3 525 = 3 × 5 2 × 7

 STEP 2: List the common prime divisors (factors) with the least power of all the given integers.  375 = 3 × 5 3 = 3 × 5 2 × 5 525 = 3 × 5 2 × 7 = 3 × 5 2 × 7  Common Prime Divisors (Factors) with Least Power: 3 and 5 2

 STEP 3: Multiply the common prime divisors (factors) to find the greatest common divisor (factor).  3 × 5 2 = 75  GCD (GCF) of 375 and 525 = 75

 The least common multiple (denominator) of two or more integers can be obtained in three steps:  STEP 1: Find the prime factorization of each integer. (Use Division Method)  4 = 2 2 10 = 2 × 5 45 = 3 2 × 5

 STEP 2: List the prime divisors (factors) with the greatest power of all the given integers.  4 = 2 2 10 = 2 × 5 45 = 3 2 × 5  Prime Divisors (Factors) with Greatest Power: 2 2, 3 2, and 5

 STEP 3: Multiply the prime divisors (factors) to find the least common multiple (denominator).  2 2 × 3 2 × 5 = 180  LCM of 4, 10 and 45 = 180 

 Find the GCD of 63 and 105. 63 = 3 2 7 105 = 3 5 7  Smallest exponent of each factor: 3 and 7  So, the GCD is 3 7 = 21.

 Find the LCM of 63 and 105. 63 = 3 2 7 105 = 3 5 7  Greatest exponent of each factor: 3 2, 5 and 7  So, the LCM is 3 2 5 7 = 315.

 Find the GCD and LCM of 48 and 54.  Prime factorizations of each: 48 = 2 2 2 2 3 = 2 4 3 54 = 2 3 3 3 = 2 3 3 GCD = 2 3 = 6 LCM = 2 4 3 3 = 432

 Evaluate: a) (3)(  4)b) (  7)(  5) c) 8 7d) (  5)(8)  Solution: a) (3)(  4) =  12b) (  7)(  5) = 35 c) 8 7 = 56d) (  5)(8) =  40

 Evaluate: a) b) c) d)  Solution: a) b) c) d)

 Fractions are numbers such as:  The numerator is the number above the fraction line.  The denominator is the number below the fraction line.

 Convert to an improper fraction.

 Convert to a mixed number.  The mixed number is

 Evaluate the following. a) b)

 Evaluate the following. a) b)

 Evaluate:  Solution:

 The principal (or positive) square root of a number n, written is the positive number that when multiplied by itself, gives n.  For example,

 Simplify: a) b)

 Simplify:

 Move the decimal point in the original number to the right or left until you obtain a number greater than or equal to 1 and less than 10.

 Write each number in scientific notation. a)1,265,000,000. 1.265  10 9 Decimal Point to left is b)0.000000000432 4.32  10  10 Decimal Point to right is

 Write each number in decimal notation. a)4.67  10 5 467,000 b)1.45  10 –7 0.000000145

 Large Number move to left and is a number  Small Number move to right and is a number

 Addition a + b = b + a for any real numbers a and b.  Multiplication a b = b a for any real numbers a and b.

 8 + 12 = 12 + 8 is a true statement.  5  9 = 9  5 is a true statement.  Note: The commutative property does not hold true for subtraction or division.

 Addition (a + b) + c = a + (b + c), for any real numbers a, b, and c.  Multiplication (a b) c = a (b c), for any real numbers a, b, and c.

 (3 + 5) + 6 = 3 + (5 + 6) is true.  (4  6)  2 = 4  (6  2) is true.  Note: The associative property does not hold true for subtraction or division.

 Distributive property of multiplication over addition a (b + c) = a b + a c for any real numbers a, b, and c.  Example: 6 (r + 12) = 6 r + 6 12 = 6r + 72

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