What Times What Equals 147

What Times What Equals 147? Unlocking the Mysteries of Factor Pairs

Finding the numbers that multiply to equal 147 might seem like a simple math problem, but it opens a door to understanding fundamental concepts in number theory, including prime factorization, factors, and multiples. This exploration goes beyond simply providing the answer; we'll delve into the methods for finding these numbers and the underlying mathematical principles. This will equip you with skills applicable to more complex multiplication problems and strengthen your understanding of number relationships.

Introduction: Exploring the World of Factors and Multiples

The question "What times what equals 147?" is essentially asking for the factor pairs of 147. A factor is a number that divides evenly into another number without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12, because each of these numbers divides evenly into 12. A multiple, conversely, is the result of multiplying a number by an integer. Multiples of 3, for example, are 3, 6, 9, 12, and so on. Finding the factor pairs of 147 means identifying all pairs of numbers whose product is 147.

Method 1: Systematic Trial and Error

The simplest approach to finding the factors of 147 is through systematic trial and error. We start by checking small numbers to see if they divide evenly into 147.

• 1: 147 divided by 1 is 147, giving us the factor pair (1, 147).
• 3: 147 divided by 3 is 49, providing the factor pair (3, 49).
• 7: 147 divided by 7 is 21, yielding the factor pair (7, 21).
• Other Numbers: Trying other numbers, we find that none divide evenly into 147.

This method reveals that the factor pairs of 147 are (1, 147), (3, 49), and (7, 21). This approach is effective for smaller numbers but becomes less efficient as the numbers get larger.

Method 2: Prime Factorization – The Key to Understanding

A more powerful and efficient method involves prime factorization. A prime number is a whole number greater than 1 that has only two divisors: 1 and itself (e.g., 2, 3, 5, 7, 11, etc.). Prime factorization is the process of expressing a number as the product of its prime factors.

To find the prime factorization of 147, we start by dividing by the smallest prime number, 2. Since 147 is odd, it's not divisible by 2. We move to the next prime number, 3.

147 ÷ 3 = 49

Now we need to factor 49. 49 is not divisible by 3 or 5, but it is divisible by 7.

49 ÷ 7 = 7

Since 7 is a prime number, we have found the prime factorization of 147: 3 x 7 x 7, or 3 x 7².

This prime factorization is unique to 147. Understanding this allows us to systematically find all factor pairs. We can combine these prime factors in different ways to generate the factor pairs:

• 3 x 49
• 7 x 21
• 1 x 147
• 1 x 3 x 49 = 147
• 1 x 7 x 21 = 147

Therefore, all possible ways to multiply to achieve 147 are: 3 x 49; 7 x 21; and 1 x 147.

Method 3: Using a Factor Tree

A visual tool to aid in prime factorization is a factor tree. Here's how to construct one for 147:

      147
     /   \
    3    49
       /  \
      7    7

The factor tree shows the stepwise breakdown of 147 into its prime factors (3, 7, and 7). From this tree, we can easily derive all the factor pairs.

Understanding the Significance of Prime Factorization

The prime factorization of a number is fundamental in various areas of mathematics. It's crucial for:

• Simplifying Fractions: Finding the greatest common divisor (GCD) of the numerator and denominator of a fraction relies heavily on prime factorization.
• Solving Equations: Prime factorization can be instrumental in solving certain types of equations, particularly those involving integers.
• Cryptography: Prime numbers are the cornerstone of many modern cryptographic systems, ensuring secure online transactions and communication.
• Number Theory: Prime factorization is a core concept in number theory, a branch of mathematics dedicated to studying the properties of integers.

Beyond 147: Applying the Methods to Other Numbers

The methods outlined above—trial and error, prime factorization, and factor trees—can be applied to find the factor pairs of any number. Let's consider finding the factors of 360:

1. Trial and Error: This becomes tedious for a larger number like 360.
2. Prime Factorization: We can find the prime factors: 360 = 2³ x 3² x 5.
3. Factor Tree:

       360
      /   \
     2    180
         /  \
        2   90
            / \
           2   45
               / \
              3   15
                  / \
                 3   5

This factor tree provides the prime factorization 2³ x 3² x 5. From this, we can systematically generate all the factor pairs of 360.

Frequently Asked Questions (FAQs)

• Q: Are there any negative factors of 147?

  A: Yes, since (-3) x (-49) = 147, (-7) x (-21) = 147, and (-1) x (-147) = 147, the negative counterparts of each factor pair are also valid factors.

• Q: How many factors does 147 have in total?

  A: 147 has six factors: 1, 3, 7, 21, 49, and 147. If we include negative factors, there are twelve in total.

• Q: What if the number is very large?

  A: For very large numbers, specialized algorithms are used to find prime factorization efficiently. These algorithms are significantly more complex than the methods described here.

• Q: Is there a formula to find the number of factors?

  A: Yes. If the prime factorization of a number N is given by N = p₁^a₁ x p₂^a₂ x ... x pₙ^aₙ, then the number of factors of N is (a₁ + 1)(a₂ + 1)...(aₙ + 1). For 147 (3 x 7²), the number of factors is (1 + 1)(2 + 1) = 6.

Conclusion: Mastering the Fundamentals of Number Theory

Finding what numbers multiply to equal 147 is more than just a simple arithmetic exercise. It provides a practical application for understanding core concepts in number theory, including prime factorization, factors, and multiples. Mastering these concepts builds a strong foundation for more advanced mathematical studies and problem-solving skills. By applying the methods described above—trial and error, prime factorization, and the factor tree—you gain the ability to tackle similar problems efficiently and confidently, regardless of the size of the number. The journey of discovering the factors of 147 unveils a fascinating glimpse into the elegance and logic that underpin the world of numbers.