Uncover All the Factors of 615: Step-by-Step Guide

Prime Factorization with Composite Numbers

Write all the factors of the following number: 615 615

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Find all prime factors of the number
00:05 The ones digit is 5, therefore 5 is definitely a prime factor
00:09 Divide by 5, and continue with the result to find factors
00:16 The ones digit is now 3, therefore 3 is definitely a prime factor
00:20 Divide by 3, and continue with the result to find factors
00:24 And the result is a prime number, so it's a factor by itself
00:27 And this is the solution to the question

Step-by-step written solution

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Understand the problem

Write all the factors of the following number: 615 615

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Step-by-step solution

To find all the factors of the number 615 615 , we'll follow these steps:

  • Step 1: Start by performing prime factorization of 615 615 .
  • Step 2: Determine if 615 615 is divisible by small prime numbers.
  • Step 3: Write down all prime factors.

Let's proceed with the solution:

Step 1: Check divisibility by small primes.
Firstly, check for divisibility by 2 2 . Since 615 615 is odd, it's not divisible by 2 2 .

Step 2: Check divisibility by 3 3 . Add the digits of 615 615 (i.e., 6+1+5=12 6 + 1 + 5 = 12 ), which is divisible by 3 3 . Thus, 615÷3=205 615 \div 3 = 205 .

Step 3: Check 205 205 for divisibility by 5 5 . Because 205 205 ends in 5 5 , it is divisible by 5 5 . So, 205÷5=41 205 \div 5 = 41 .

Step 4: Check 41 41 . This is a prime number.

Thus, the prime factorization of 615 615 is 3×5×41 3 \times 5 \times 41 .

Step 5: Now, list all the prime factors of 615 615 , which are 3 3 , 5 5 , and 41 41 .

Therefore, the factors of 615 615 include exactly these prime divisors.

Hence, the solution to the problem is:

3,5,41 3, 5, 41 .

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Final Answer

5,3,41 5,3,41

Key Points to Remember

Essential concepts to master this topic
  • Prime Factorization: Break down numbers into their prime building blocks
  • Technique: Test divisibility: 615÷3=205 615 ÷ 3 = 205 , then 205÷5=41 205 ÷ 5 = 41
  • Check: Multiply prime factors back: 3×5×41=615 3 × 5 × 41 = 615

Common Mistakes

Avoid these frequent errors
  • Confusing prime factors with all factors
    Don't list only prime factors when asked for all factors = incomplete answer! The question asks for ALL factors, not just prime ones. Always find prime factors first, then use them to generate ALL possible factors including 1, the number itself, and composite factors.

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?

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Prime factors are the basic building blocks (like 3, 5, 41 for 615). All factors include every number that divides evenly: 1, 3, 5, 15, 41, 123, 205, and 615!

How do I find ALL factors once I have prime factors?

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Use the prime factors to build combinations! For 615=3×5×41 615 = 3 × 5 × 41 , multiply different combinations: 3×5=15, 3×41=123, 5×41=205, plus 1 and 615 itself.

Why does the sum of digits test work for divisibility by 3?

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It's a special rule! If the sum of digits is divisible by 3, the whole number is too. For 615: 6+1+5=12, and since 12÷3=4, we know 615 is divisible by 3.

How do I know when I've found all the prime factors?

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Keep dividing until you reach a prime number that can't be divided further. Like 41 - test if smaller primes divide it. Since none do, 41 is prime and you're done!

What if I can't tell if a number is prime?

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Test divisibility by primes up to the square root! For 41, test primes up to √41 ≈ 6.4, so check 2, 3, 5. Since none divide evenly, 41 is prime.

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