Identifying the Prime Factor in a Three-Digit Number ?12

Q: Why is 2 the only guaranteed prime factor?

Because any number ending in 2 is automatically divisible by 2! Whether it's 112, 312, 512, or 912 - they all have 2 as a factor since they end in an even digit.

Q: What about the other numbers like 3, 7, or 11?

These might be factors for some numbers ending in 12, but not all of them. For example, 112 ÷ 7 = 16 (so 7 is a factor), but 212 ÷ 7 = 30.28... (so 7 is not a factor).

Q: How do I check if a number is divisible by 2?

Super easy! Just look at the last digit. If it's 0, 2, 4, 6, or 8 (even numbers), then the whole number is divisible by 2.

Q: What does 'first factors' mean?

It means the prime factors when you break down the number. For example, 12 = 2 × 2 × 3, so the prime factors are 2 and 3. Since any ?12 number is even, 2 will always appear in its prime factorization.

Divisibility Rules with Prime Factorization

I am a three-digit number $?12$

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 2, therefore the number is even

00:06 Every even number is divisible by 2

00:09 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 $?12$

Which prime factor will surely appear among my first factors?

Step-by-step solution

To solve this problem, we will make use of divisibility rules, particularly focusing on the rule for 2.

Step 1: Analyze the given number form, $?12$ , which indicates the number is a three-digit integer ending with the digits 12.
Step 2: Apply divisibility rules. Note the last digit of $12$ is an even number, which is 2. By the rule of divisibility by 2, any number ending in an even number is divisible by 2.
Step 3: Since the last digit is 2, it confirms the number is divisible by 2. Therefore, 2 is a prime factor of the number $?12$ .

Given the options, $2$ is the only prime factor that will certainly appear among the first factors of any number ending with 12, as other numbers such as 3, 7, or 11 do not have guaranteed divisibility given non-fixed sum of digits or specific rules not directly applicable.

Therefore, the solution to the problem is $2$ .

Final Answer

$2$

Key Points to Remember

Essential concepts to master this topic

Rule: Numbers ending in even digits are always divisible by 2
Technique: Check last digit - 312, 412, 512 all end in 2
Check: Any three-digit ?12 number divided by 2 gives whole number ✓

Common Mistakes

Avoid these frequent errors

Testing other factors without checking divisibility rules first
Don't check if 312 is divisible by 3, 7, or 11 without using divisibility rules = wasted time and wrong focus! These factors may or may not appear. Always use the divisibility rule for 2 first - any number ending in an even digit is guaranteed divisible by 2.

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 is 2 the only guaranteed prime factor?

Because any number ending in 2 is automatically divisible by 2! Whether it's 112, 312, 512, or 912 - they all have 2 as a factor since they end in an even digit.

What about the other numbers like 3, 7, or 11?

These might be factors for some numbers ending in 12, but not all of them. For example, 112 ÷ 7 = 16 (so 7 is a factor), but 212 ÷ 7 = 30.28... (so 7 is not a factor).

How do I check if a number is divisible by 2?

Super easy! Just look at the last digit. If it's 0, 2, 4, 6, or 8 (even numbers), then the whole number is divisible by 2.

Could there be other guaranteed prime factors?

No! Since we don't know what the first digit is (the ? could be 1, 2, 3, etc.), we can't guarantee divisibility by any other prime. Only the divisibility rule for 2 works here.

What does 'first factors' mean?

It means the prime factors when you break down the number. For example, 12 = 2 × 2 × 3, so the prime factors are 2 and 3. Since any ?12 number is even, 2 will always appear in its prime factorization.

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