Find the Number with Prime Factors 5, 7, and 5: Mathematical Challenge

Q: What does it mean when a prime factor appears twice?

When a prime factor repeats, like the two 5's, it means that prime divides the number multiple times. Think of it as $ 5^2 $ (5 squared) in the prime factorization.

Q: Why do I multiply instead of add the factors?

Prime factorization shows multiplication, not addition! The factors 5, 7, and 5 mean 5 times 7 times 5. Adding would give you a completely different number.

Q: How can I check if 175 is correct?

Divide 175 by each prime factor: $ 175 ÷ 5 = 35 $, then $ 35 ÷ 5 = 7 $, then $ 7 ÷ 7 = 1 $. If you get 1 at the end, you're right!

Q: Could there be other numbers with these same factors?

No! Prime factorization is unique. Only one number has exactly the prime factors 5, 7, and 5. This is called the Fundamental Theorem of Arithmetic.

Prime Factorization with Repeated Factors

What is the number whose prime factors are: $5,7,5$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the number with the given prime factors

00:03 To find the number, multiply all factors one by one

00:08 Calculate one multiplication at a time and continue

00:13 Break down 35 into 30 plus 5, and multiply accordingly

00:30 Calculate the multiplications

00:35 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

What is the number whose prime factors are: $5,7,5$

Step-by-step solution

To solve this problem, we'll multiply the given prime factors $5, 7,$ and $5$ to determine the original number.

Step 1: Multiply the prime factors together: $(5 \times 5) \times 7$ .
Step 2: Calculate $5 \times 5$ , which equals $25$ .
Step 3: Multiply the result by the remaining prime factor: $25 \times 7 = 175$ .

Therefore, the solution to the problem is $175$ .

Final Answer

$175$

Key Points to Remember

Essential concepts to master this topic

Rule: Multiply all prime factors together to find the number
Technique: Group repeated factors: $5 \times 5 = 25$ , then multiply by 7
Check: Verify by factoring 175: $175 = 5^2 \times 7$ ✓

Common Mistakes

Avoid these frequent errors

Adding prime factors instead of multiplying
Don't add 5 + 7 + 5 = 17! This gives a completely wrong number that doesn't match the prime factorization. Always multiply all prime factors together: 5 × 5 × 7 = 175.

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 does it mean when a prime factor appears twice?

When a prime factor repeats, like the two 5's, it means that prime divides the number multiple times. Think of it as $5^2$ (5 squared) in the prime factorization.

Why do I multiply instead of add the factors?

Prime factorization shows multiplication, not addition! The factors 5, 7, and 5 mean 5 times 7 times 5. Adding would give you a completely different number.

How can I check if 175 is correct?

Divide 175 by each prime factor: $175 ÷ 5 = 35$ , then $35 ÷ 5 = 7$ , then $7 ÷ 7 = 1$ . If you get 1 at the end, you're right!

What if the prime factors were in a different order?

Order doesn't matter in multiplication! Whether you write 5, 7, 5 or 7, 5, 5 or 5, 5, 7, the answer is still the same: 175.

Could there be other numbers with these same factors?

No! Prime factorization is unique. Only one number has exactly the prime factors 5, 7, and 5. This is called the Fundamental Theorem of Arithmetic.

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