15 Questions
What is the unique prime factorization of 24?
2 × 2 × 2 × 3
What is the LCM of 9 and 15?
45
What is the HCF of 15 and 20?
5
Tom needs 15 boxes to pack his toys and Alex needs 20 boxes to pack his toys. What is the minimum number of boxes they need to buy to satisfy both?
60
A bookshelf has 15 books on each shelf and a library has 20 shelves. What is the greatest number of books that can be removed from the bookshelf and still leave an equal number of books on each shelf?
5
What is the prime factorization of 36?
2 × 2 × 3 × 3
What is the LCM of 12 and 15?
60
What is the HCF of 24 and 30?
6
A factory produces 12 units of a product per hour and 18 units of another product per hour. What is the minimum number of hours they need to produce to have an equal number of units of both products?
36
What is a consequence of the Fundamental Theorem of Arithmetic?
Every positive integer can be expressed as a product of prime powers.
What is a key feature of the Fundamental Theorem of Arithmetic?
The prime factorization of a positive integer is unique.
What is an application of the Fundamental Theorem of Arithmetic?
Computer science and cryptography.
What is a result of the Fundamental Theorem of Arithmetic in number theory?
The study of prime numbers and their properties.
What is a property of prime numbers in the Fundamental Theorem of Arithmetic?
Prime numbers can be expressed as a product of prime powers.
What is true about the prime factorization of a positive integer?
It is unique, except for the order in which the prime numbers are listed.
Study Notes
Real Numbers
Fundamental Theorem of Arithmetic
- Every positive integer can be expressed as a product of prime numbers in a unique way, except for the order in which the prime numbers are listed.
- This theorem provides a way to factorize any positive integer into its prime factors.
- Example: 12 = 2 × 2 × 3 (unique prime factorization)
LCM and HCF (GCD) Word Problems
LCM (Least Common Multiple)
- The smallest number that is a multiple of two or more numbers.
- Example: Find the LCM of 6 and 8.
- Multiples of 6: 6, 12, 18, 24, 30, ...
- Multiples of 8: 8, 16, 24, 32, 40, ...
- LCM = 24 (the first number that appears in both lists)
HCF (Highest Common Factor) or GCD (Greatest Common Divisor)
- The largest number that divides two or more numbers without leaving a remainder.
- Example: Find the HCF of 12 and 18.
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
- HCF = 6 (the largest number that appears in both lists)
Word Problems
- Example 1: Tom and Alex are baking cookies. Tom needs 12 cookies for his recipe and Alex needs 18 cookies for hers. What is the minimum number of cookies they need to buy to satisfy both recipes?
- LCM of 12 and 18 = 36 (they need to buy at least 36 cookies)
- Example 2: A bookshelf has 12 books on each shelf and a library has 18 shelves. What is the greatest number of books that can be removed from the bookshelf and still leave an equal number of books on each shelf?
- HCF of 12 and 18 = 6 (they can remove 6 books from each shelf)
Real Numbers
Fundamental Theorem of Arithmetic
- Every positive integer can be expressed as a product of prime numbers in a unique way, except for the order in which the prime numbers are listed.
- This theorem provides a way to factorize any positive integer into its prime factors.
- Example: 12 = 2 × 2 × 3 (unique prime factorization).
LCM and HCF (GCD)
LCM (Least Common Multiple)
- The smallest number that is a multiple of two or more numbers.
- Example: Find the LCM of 6 and 8.
- Multiples of 6: 6, 12, 18, 24, 30,...
- Multiples of 8: 8, 16, 24, 32, 40,...
- LCM = 24 (the first number that appears in both lists).
HCF (Highest Common Factor) or GCD (Greatest Common Divisor)
- The largest number that divides two or more numbers without leaving a remainder.
- Example: Find the HCF of 12 and 18.
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
- HCF = 6 (the largest number that appears in both lists).
Word Problems
- Example 1: Tom and Alex need to buy cookies in bulk.
- Tom needs 12 cookies for his recipe and Alex needs 18 cookies for hers.
- LCM of 12 and 18 = 36 (they need to buy at least 36 cookies).
- Example 2: A bookshelf has 12 books on each shelf and a library has 18 shelves.
- HCF of 12 and 18 = 6 (they can remove 6 books from each shelf).
Fundamental Theorem of Arithmetic
Definition
- Every positive integer can be expressed as a product of prime numbers in a unique way, except for the order of the prime numbers.
Key Features
- The prime factorization of a positive integer is unique, meaning it cannot be expressed as a product of different prime numbers.
- Every positive integer can be expressed as a product of prime numbers.
Examples
- 6 is expressed as 2 × 3, which is its unique prime factorization.
- 12 is expressed as 2 × 2 × 3, which is its unique prime factorization.
- 24 is expressed as 2 × 2 × 2 × 3, which is its unique prime factorization.
Importance
- The Fundamental Theorem of Arithmetic provides a foundation for many results in number theory.
- It has applications in mathematics, computer science, and cryptography.
Consequences
- Every positive integer can be expressed as a product of prime powers, where each prime power is of the form pn, where p is a prime number and n is a positive integer.
- The prime factorization of a positive integer can be used to find its greatest common divisor (GCD) and least common multiple (LCM) with other integers.
Understand the fundamental theorem of arithmetic, prime factorization, LCM, and HCF through word problems and examples.
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