What Times What Equals 85?
Finding the factors of a number, in this case 85, is a fundamental skill in mathematics. It's the process of determining which numbers multiply together to produce that specific number. In this case, we're searching for two numbers that, when multiplied, result in 85.
Prime Factorization: A Key to Finding Factors
One efficient method for finding factors is prime factorization. A prime number is a whole number greater than 1 that has only two divisors: 1 and itself. Prime factorization involves breaking down a number into its prime number components.
Let's factorize 85:
- 85 is not divisible by 2 or 3.
- 85 is divisible by 5 (85 ÷ 5 = 17)
- 17 is a prime number.
Therefore, the prime factorization of 85 is 5 x 17.
Finding Pairs of Factors
Now that we have the prime factors of 85, we can identify all its pairs of factors:
- 1 x 85 = 85
- 5 x 17 = 85
Answer: The numbers that multiply to give 85 are 1 and 85, and 5 and 17.
Applications of Finding Factors
Understanding factors has various applications in mathematics and beyond:
- Simplifying fractions: Finding the greatest common factor (GCF) of the numerator and denominator allows you to simplify fractions.
- Algebraic equations: Factorization is crucial for solving quadratic equations and other algebraic expressions.
- Real-life problem-solving: It's used in situations involving dividing quantities, sharing items equally, or determining combinations.
Example: Imagine you have 85 apples to distribute equally among a group of people. Knowing the factors of 85 allows you to determine different ways to divide the apples. You could distribute them to 1 group of 85 people, or 5 groups of 17 people, or 17 groups of 5 people.
Tips for Finding Factors
- Start with the smallest prime numbers: Begin by checking divisibility by 2, 3, 5, 7, 11, and so on.
- Use divisibility rules: There are handy divisibility rules for certain numbers (e.g., a number is divisible by 2 if it ends in an even number, a number is divisible by 5 if it ends in 0 or 5).
- Factorization calculator: For larger numbers, online calculators can assist with prime factorization and finding factors.
Conclusion
Finding the factors of a number, like 85, is a fundamental mathematical skill with diverse applications. Prime factorization is a powerful technique for determining these factors. By understanding factors, we can solve various mathematical problems and gain insights into real-world situations.