What are the factors of 250?
Understand the Problem
The question is asking us to find all the numbers that can multiply together to result in 250. This entails identifying both the prime factors and all possible factor pairs of 250.
Answer
The prime factorization of 250 is $2 \times 5^3$, and the factor pairs are $(1, 250)$, $(2, 125)$, $(5, 50)$, $(10, 25)$, $(25, 10)$, $(50, 5)$, $(125, 2)$, $(250, 1)$.
Answer for screen readers
The prime factorization of 250 is $2 \times 5^3$, and the factor pairs of 250 are $(1, 250)$, $(2, 125)$, $(5, 50)$, $(10, 25)$, $(25, 10)$, $(50, 5)$, $(125, 2)$, $(250, 1)$.
Steps to Solve
- Identify Prime Factorization
To find the prime factors of 250, we divide the number by the smallest prime number until we reach 1.
$$ 250 \div 2 = 125 $$
Next, we check 125 with the next smallest prime number:
$$ 125 \div 5 = 25 $$
Continuing with 25:
$$ 25 \div 5 = 5 $$
And finally:
$$ 5 \div 5 = 1 $$
So, the prime factorization of 250 is $2 \times 5^3$.
- List All Factor Pairs
Using the prime factors, we can determine all factor pairs of 250.
The factors of 250 are obtained by multiplying combinations of its prime factors, which gives us the pairs:
- $1 \times 250$
- $2 \times 125$
- $5 \times 50$
- $10 \times 25$
- $25 \times 10$
- $50 \times 5$
- $125 \times 2$
- $250 \times 1$
- Summarize Factor Pairs
To summarize, the complete set of factor pairs of 250 is:
- $(1, 250)$
- $(2, 125)$
- $(5, 50)$
- $(10, 25)$
- $(25, 10)$
- $(50, 5)$
- $(125, 2)$
- $(250, 1)$
The prime factorization of 250 is $2 \times 5^3$, and the factor pairs of 250 are $(1, 250)$, $(2, 125)$, $(5, 50)$, $(10, 25)$, $(25, 10)$, $(50, 5)$, $(125, 2)$, $(250, 1)$.
More Information
The number 250 is an even number and can be expressed as a product of prime numbers, which is a foundational concept in number theory. Prime factorization is useful in various areas of mathematics, including simplifying fractions and solving least common multiple problems.
Tips
- Confusing factors with multiples: Remember that factors are numbers that you multiply together to get a result, whereas multiples are the results of multiplying a number by an integer.
- Forgetting to list all factor pairs: It’s important to ensure you consider the pairs in both orders and include the inverse.
