Precisely List All Factors of 350: A Step-by-Step Guide

Q: What's the difference between prime factors and all factors?

Prime factors are the basic building blocks (like 2, 5, 7 for 350). All factors include every number that divides 350 evenly, including products of the prime factors like 10, 25, 35.

Q: How many factors should 350 have?

Use the formula: If $ n = p_1^{a_1} \times p_2^{a_2} \times p_3^{a_3} $, then the number of factors is $ (a_1+1)(a_2+1)(a_3+1) $. For 350 = $ 2^1 \times 5^2 \times 7^1 $, that's (1+1)(2+1)(1+1) = 12 factors.

Q: Do I need to include 1 and 350 as factors?

Yes! Every number has 1 and itself as factors. The number 1 divides everything, and every number divides itself exactly once. These are called trivial factors.

Q: What if I miss a factor in my list?

Double-check by dividing 350 by each factor you found. If the result is also in your list, you're on track! For example: 350 ÷ 14 = 25, and both 14 and 25 should be in your factor list.

Q: Can factors be larger than the square root of the number?

Absolutely! Factors come in pairs. If $ a \times b = 350 $, then both a and b are factors. One will be ≤ √350 ≈ 18.7, and the other will be ≥ 18.7.

Q: Is there a pattern to finding factors systematically?

Yes! Start with the prime factorization, then use each prime factor raised to every possible power from 0 to its maximum. For $ 2^1 \times 5^2 \times 7^1 $, combine: $ 2^{0,1} \times 5^{0,1,2} \times 7^{0,1} $.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find all the prime factors of the number

00:03 The ones digit is 0, therefore 2 is definitely a prime factor

00:08 Divide by 2, and continue with the result to find the factors

00:14 The ones digit is 5, therefore 5 is definitely a prime factor

00:22 Divide by 5, and continue with the result to find the factors

00:29 The ones digit is 5, therefore 5 is definitely a prime factor

00:33 Divide by 5, and continue with the result to find the factors

00:37 And the result is a prime number, therefore it is a factor by itself

00:40 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Write all the factors of the following number: $350$

2

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Perform the prime factorization of $350$ .
Step 2: Identify all possible combinations of the factors.
Step 3: Verify each step to ensure all factors are included.

Now, let's work through each step:
Step 1: Perform the prime factorization of $350$
The number $350$ is even, and hence divisible by $2$ :
$350 \div 2 = 175$
Next, $175$ ends in $5$ , indicating it is divisible by $5$ :
$175 \div 5 = 35$
Next, $35$ is also divisible by $5$ (it ends in $5$ ):
$35 \div 5 = 7$
Finally, $7$ is a prime number.
Thus, the prime factorization of $350$ is $2 \times 5^2 \times 7$ .

Step 2: Identify all possible combinations of factors.
Using the prime factors, combine them to list all factors of $350$ :
- $1$ (trivial factor)
- $2$
- $5$
- $7$
- $2 \times 5 = 10$
- $2 \times 7 = 14$
- $5 \times 5 = 25$
- $5 \times 7 = 35$
- $2 \times 5^2 = 50$
- $2 \times 5 \times 7 = 70$
- $5^2 \times 7 = 175$
- $350$ (the number itself)

Step 3: Verify each step.
Verify each combination using basic multiplication to ensure accuracy.

Therefore, all the factors of $350$ are:
$1, 2, 5, 7, 10, 14, 25, 35, 50, 70, 175, 350$ .

3

Final Answer

$2,5,7,5$

Key Points to Remember

Essential concepts to master this topic

Rule: Use prime factorization to find all factors systematically
Technique: For $350 = 2 \times 5^2 \times 7$ , combine powers: 1, 2, 5, 7, 10, 14, 25, 35, 50, 70, 175, 350
Check: Each factor divides 350 evenly: 350 ÷ 25 = 14 ✓

Common Mistakes

Avoid these frequent errors

Confusing prime factors with all factors
Don't list only prime factors like 2, 5, 7 = incomplete answer! This misses compound factors like 10, 25, 35, etc. Always find prime factorization first, then combine all possible products of those primes.

Practice Quiz

Test your knowledge with interactive questions

Write all the factors of the following number: $$ 6 $$

FAQ

Everything you need to know about this question

What's the difference between prime factors and all factors?

+

Prime factors are the basic building blocks (like 2, 5, 7 for 350). All factors include every number that divides 350 evenly, including products of the prime factors like 10, 25, 35.

How many factors should 350 have?

+

Use the formula: If $n = p_1^{a_1} \times p_2^{a_2} \times p_3^{a_3}$ , then the number of factors is $(a_1+1)(a_2+1)(a_3+1)$ . For 350 = $2^1 \times 5^2 \times 7^1$ , that's (1+1)(2+1)(1+1) = 12 factors.

Do I need to include 1 and 350 as factors?

+

Yes! Every number has 1 and itself as factors. The number 1 divides everything, and every number divides itself exactly once. These are called trivial factors.

What if I miss a factor in my list?

+

Double-check by dividing 350 by each factor you found. If the result is also in your list, you're on track! For example: 350 ÷ 14 = 25, and both 14 and 25 should be in your factor list.

Can factors be larger than the square root of the number?

+

Absolutely! Factors come in pairs. If $a \times b = 350$ , then both a and b are factors. One will be ≤ √350 ≈ 18.7, and the other will be ≥ 18.7.

Is there a pattern to finding factors systematically?

+

Yes! Start with the prime factorization, then use each prime factor raised to every possible power from 0 to its maximum. For $2^1 \times 5^2 \times 7^1$ , combine: $2^{0,1} \times 5^{0,1,2} \times 7^{0,1}$ .

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Precisely List All Factors of 350: A Step-by-Step Guide

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