Which Prime Factor Lurks in the Two-Digit Number ?4

Q: Why is 2 always a factor of numbers ending in 4?

Because any number ending in an even digit (0, 2, 4, 6, 8) is divisible by 2! Since 4 is even, every number ending in 4 is automatically even.

Q: What about the other answer choices like 3, 5, or 7?

These might be factors of some numbers ending in 4, but not all of them. For example: 14 ÷ 7 = 2, but 24 ÷ 7 = 3.43... Only 2 divides every ?4 number.

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

Look at the last digit only! If it's 0, 2, 4, 6, or 8, the entire number is divisible by 2. No need to do the actual division.

Q: Are there other guaranteed prime factors for ?4 numbers?

No, only 2 is guaranteed for all ?4 numbers. Other prime factors depend on the tens digit. For example, 30 has factors 2, 3, and 5, but 32 only has factor 2.

Q: What does 'surely appear' mean in this problem?

It means the prime factor appears in every single number of the form ?4, not just some of them. That's why 2 is the answer - it's in 14, 24, 34, 44, 54, 64, 74, 84, and 94!

Prime Factorization with Two-Digit Patterns

I am a two-digit number $?4$

Which prime factor will surely appear among my first factors?

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

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?

00:04 The ones digit is 4, therefore the number is even

00:07 Every even number is divisible by 2

00:10 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 two-digit number $?4$

Which prime factor will surely appear among my first factors?

Step-by-step solution

To solve this problem, let's analyze the numbers:

Step 1: Identify that all numbers of the form $?4$ end with the digit 4, which is even.
Step 2: Apply the divisibility rule for 2: A number ending in an even digit is divisible by 2.
Step 3: Recognize that since all such numbers are even due to ending in 4, each number is divisible by 2.

Therefore, every number ending in 4 has 2 as a factor.

To further clarify, consider some examples: 14, 24, 34, ..., 94. These can all be divided by 2.

Therefore, the prime factor that will surely appear among the factors of any number in the form $"?4"$ is $2$ .

Final Answer

$2$

Key Points to Remember

Essential concepts to master this topic

Even Numbers: Any number ending in 4 is even and divisible by 2
Pattern Recognition: Numbers like 14, 24, 34 all have factor 2
Verification: Check any ?4 number divides evenly by 2 ✓

Common Mistakes

Avoid these frequent errors

Checking individual numbers instead of recognizing the pattern
Don't test each number like 14, 24, 34 separately = wasting time and missing the pattern! This approach doesn't reveal why ALL ?4 numbers share the same prime factor. Always recognize that ending in 4 makes every number even, so 2 is always a factor.

Practice Quiz

Test your knowledge with interactive questions

Write all the factors of the following number: $$ 9 $$

FAQ

Everything you need to know about this question

Why is 2 always a factor of numbers ending in 4?

Because any number ending in an even digit (0, 2, 4, 6, 8) is divisible by 2! Since 4 is even, every number ending in 4 is automatically even.

What about the other answer choices like 3, 5, or 7?

These might be factors of some numbers ending in 4, but not all of them. For example: 14 ÷ 7 = 2, but 24 ÷ 7 = 3.43... Only 2 divides every ?4 number.

How can I quickly check if a number is divisible by 2?

Look at the last digit only! If it's 0, 2, 4, 6, or 8, the entire number is divisible by 2. No need to do the actual division.

Are there other guaranteed prime factors for ?4 numbers?

No, only 2 is guaranteed for all ?4 numbers. Other prime factors depend on the tens digit. For example, 30 has factors 2, 3, and 5, but 32 only has factor 2.

What does 'surely appear' mean in this problem?

It means the prime factor appears in every single number of the form ?4, not just some of them. That's why 2 is the answer - it's in 14, 24, 34, 44, 54, 64, 74, 84, and 94!

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Division - Advanced questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime