Problem 75
Question
According to the Fundamental Theorem of Arithmetic, every natural number greater than 1 can be written as the product of primes in a unique way, except for the order of the factors. For example, \(45=3 \cdot 3 \cdot 5\). Write each of the following as a product of primes. (a) 243 (b) 124 (c) 5100
Step-by-Step Solution
Verified Answer
(a) 243 = \(3^5\); (b) 124 = \(2^2 \cdot 31\); (c) 5100 = \(2^2 \cdot 3 \cdot 5^2 \cdot 17\).
1Step 1: Understand the Problem
We need to decompose each given number into its prime factors. This means we'll find the prime numbers that multiply together to make each of these original numbers.
2Step 2: Factorization of 243
Start with number 243. Check for the smallest prime number, which is 2. Since 243 is odd, it is not divisible by 2. Try the next prime, 3: \\[\frac{243}{3} = 81\] \Repeat the process: \\[\frac{81}{3} = 27, \quad \frac{27}{3} = 9, \quad \frac{9}{3} = 3, \quad \frac{3}{3} = 1\] \This shows that 243 is \(3^5\).
3Step 3: Factorization of 124
Start with number 124. It is even, so divide by 2: \\[\frac{124}{2} = 62\] \Again 62 is even: \\[\frac{62}{2} = 31\] \31 is a prime number. Hence, 124 can be factored as \(2^2 \cdot 31\).
4Step 4: Factorization of 5100
Begin with 5100. It's even, so divide by 2 repeatedly: \\[\frac{5100}{2} = 2550, \quad \frac{2550}{2} = 1275\] \1275 is not even, check divisibility by 3: \\[\frac{1275}{3} = 425\] \425 ends in 5, divide by 5 repeatedly: \\[\frac{425}{5} = 85, \quad \frac{85}{5} = 17\] \17 is a prime number. Therefore, 5100 is \(2^2 \cdot 3 \cdot 5^2 \cdot 17\).
Key Concepts
Prime FactorizationPrime NumbersNatural NumbersFactorization Steps
Prime Factorization
Prime factorization is the process of breaking down a natural number into a product of prime numbers. This is a fundamental concept in mathematics, rooted in the Fundamental Theorem of Arithmetic.
According to this theorem, every natural number greater than 1 can be expressed in exactly one way as a product of prime numbers, aside from the order of these primes.
For example, the prime factorization of 45 is \(3 \cdot 3 \cdot 5\), or \(3^2 \cdot 5\). This is unique, meaning no other combination of prime numbers will multiply to give 45.
According to this theorem, every natural number greater than 1 can be expressed in exactly one way as a product of prime numbers, aside from the order of these primes.
For example, the prime factorization of 45 is \(3 \cdot 3 \cdot 5\), or \(3^2 \cdot 5\). This is unique, meaning no other combination of prime numbers will multiply to give 45.
- Unique: Every number >1 has one set of prime factors.
- Order doesn't matter: \(3 \cdot 5 \cdot 3\) is the same as \(3^2 \cdot 5\).
Prime Numbers
Prime numbers are the building blocks of all natural numbers. They are numbers greater than 1 that have no divisors other than 1 and themselves.
Examples include 2, 3, 5, and 7. Unlike composite numbers, prime numbers cannot be divided evenly by any other numbers.
This property makes primes the key components of prime factorization.
Examples include 2, 3, 5, and 7. Unlike composite numbers, prime numbers cannot be divided evenly by any other numbers.
This property makes primes the key components of prime factorization.
- Prime numbers must be greater than 1.
- They can only be divided by 1 and themselves.
- They serve as the 'building blocks' in number theory.
Natural Numbers
Natural numbers are the numbers we typically use for counting. They start from 1 and continue upwards (1, 2, 3, ...).
In the context of prime factorization, these are the numbers we are interested in factoring into prime numbers. Every natural number greater than 1 is involved in the Fundamental Theorem of Arithmetic, ensuring each has its unique prime factorization.
Even though our daily use of numbers often includes zero, zero is not a natural number.
In the context of prime factorization, these are the numbers we are interested in factoring into prime numbers. Every natural number greater than 1 is involved in the Fundamental Theorem of Arithmetic, ensuring each has its unique prime factorization.
Even though our daily use of numbers often includes zero, zero is not a natural number.
- Natural numbers start from 1 upwards.
- They do not include fractions or decimals.
- All natural numbers greater than 1 can be factored into primes.
Factorization Steps
To factor a number into its prime components, follow a series of logical steps. These steps ensure we find the prime factors accurately.
Begin by dividing the number by the smallest prime number possible, typically starting with 2. If the number is odd, move to the next prime, which is 3, and continue checking divisibility by increasing primes (5, 7, 11, ...).
Continue this division process until the result is 1, meaning all factors have been identified. Each successful division reveals a prime factor.
Begin by dividing the number by the smallest prime number possible, typically starting with 2. If the number is odd, move to the next prime, which is 3, and continue checking divisibility by increasing primes (5, 7, 11, ...).
Continue this division process until the result is 1, meaning all factors have been identified. Each successful division reveals a prime factor.
- Start with the smallest prime, often 2.
- If the division isn't even, move to the next prime.
- Repeat until you reach 1 in division results.
Other exercises in this chapter
Problem 75
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Use the Fundamental Theorem of Arithmetic (Problem 75) to show that the square of any natural number greater than 1 can be written as the product of primes in a
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