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Published byGervase Holland Modified over 8 years ago

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Number Theory

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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.

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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.

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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

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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

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

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

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Convert to an improper fraction.

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Convert to a mixed number. The mixed number is

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Evaluate the following. a) b)

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Evaluate the following. a) b)

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Add: Subtract:

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Evaluate: Solution:

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

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Simplify: a) b)

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Simplify:

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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.

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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

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

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

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

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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.

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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.

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(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.

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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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