Identify the Mandatory Prime Factor in the Three-Digit Number 3?0

Q: Why does ending in 0 guarantee the number has 5 as a factor?

Any number ending in 0 is divisible by 10. Since $ 10 = 2 \times 5 $, every such number must contain both 2 and 5 as prime factors.

Q: What about the 3 at the beginning - isn't that always a factor?

Not necessarily! For example, 320 starts with 3 but $ 320 = 2^6 \times 5 $ - no factor of 3. The starting digit doesn't guarantee divisibility.

Q: How do I remember which prime factors come from ending digits?

Ending in 0: divisible by 10 = factors of 2 and 5Ending in 2, 4, 6, 8: divisible by 2Ending in 5: divisible by 5

Q: What if the middle digit changes the prime factors?

The middle digit might add additional prime factors, but it cannot remove the guaranteed ones. Since 3?0 always ends in 0, it will always have 2 and 5 as factors.

Prime Factorization with Ending Digits

I am a three-digit number $3?0$

Which prime factor will surely appear among my first factors?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Which factor definitely appears in the prime factors of the number?

00:03 The ones digit is 0

00:07 Let's try dividing various tens by factor 5 and see that it divides

00:17 Actually, any number with tens and ones digit of 0 is divisible by 5

00:21 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

I am a three-digit number $3?0$

Which prime factor will surely appear among my first factors?

Step-by-step solution

To solve this problem, let's identify the prime factor that is certain to be a part of a number in the format $3?0$ . This number always ends with 0, indicating it is divisible by 10.

Step-by-step Solution:

The number ends in 0: $3?0$ . This means it is divisible by 10.
Since a number divisible by 10 contains the prime factors of 10, which are 2 and 5, it means both 2 and 5 are factors of any such number.
Out of the given prime factor options, 3, 5, 11, and 7, the prime factor 5 is certain to be present because every such number must end in 0, confirming divisibility by 5.

Thus, the prime factor that will surely appear among the first factors of a three-digit number in the format $3?0$ is $\mathbf{5}$ .

Final Answer

$5$

Key Points to Remember

Essential concepts to master this topic

Rule: Numbers ending in 0 are always divisible by 10
Technique: Factor 10 into primes: 10 = 2 × 5
Check: Any number 3?0 contains both 2 and 5 as factors ✓

Common Mistakes

Avoid these frequent errors

Assuming the first digit determines all prime factors
Don't think that 3?0 must have 3 as a factor because it starts with 3 = wrong conclusion! The first digit doesn't guarantee divisibility. Always focus on what the ending digit tells you about guaranteed 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

Why does ending in 0 guarantee the number has 5 as a factor?

Any number ending in 0 is divisible by 10. Since $10 = 2 \times 5$ , every such number must contain both 2 and 5 as prime factors.

What about the 3 at the beginning - isn't that always a factor?

Not necessarily! For example, 320 starts with 3 but $320 = 2^6 \times 5$ - no factor of 3. The starting digit doesn't guarantee divisibility.

Could 11 or 7 also be guaranteed factors?

No! Numbers like 310, 320, 340, etc. don't necessarily contain 11 or 7 as factors. Only the ending digit 0 guarantees specific prime factors.

How do I remember which prime factors come from ending digits?

Ending in 0: divisible by 10 = factors of 2 and 5
Ending in 2, 4, 6, 8: divisible by 2
Ending in 5: divisible by 5

What if the middle digit changes the prime factors?

The middle digit might add additional prime factors, but it cannot remove the guaranteed ones. Since 3?0 always ends in 0, it will always have 2 and 5 as factors.

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