Identifying Prime Factors: The Sure Inclusion in Two-Digit Numbers

Q: Why is 2 always a factor of numbers ending in 0?

Any number ending in 0 is automatically even (divisible by 2). Since 2 is prime, it will always appear in the prime factorization of these numbers.

Q: What about 5? Doesn't 10 = 2 × 5?

While it's true that $ 10 = 2 \times 5 $, the question asks about factors that surely appear. Some ?0 numbers might have 5 grouped differently in their complete factorization, but 2 is always guaranteed.

Q: Could 6 be a prime factor?

No! The number 6 is not prime because $ 6 = 2 \times 3 $. Only numbers with exactly two factors (1 and themselves) are prime.

Q: How do I identify prime factors quickly?

Start with the smallest prime numbers: 2, 3, 5, 7, 11... Check if your number is divisible by each one. For ?0 numbers, you can immediately write down 2 as a factor!

Q: What if the number is 10 itself?

Perfect example! $ 10 = 2 \times 5 $, so both 2 and 5 are prime factors. But across all possible ?0 numbers (10, 20, 30, 40...), only 2 appears in every single one.

Prime Factorization with Two-Digit Endings

I am a two-digit number $?0$

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?

00:04 The ones digit is 0, therefore the number is even

00:07 Every even number is divisible by 2

00:13 Actually, any number with tens digit and ones digit being 0 is divisible by 5

00:20 And this is the solution to the question

Step-by-step written solution

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Understand the problem

I am a two-digit number $?0$

Which prime factor will surely appear among my first factors?

Final Answer

$2$

Key Points to Remember

Essential concepts to master this topic

Pattern Recognition: All two-digit numbers ending in 0 are multiples of 10
Technique: Factor 10 first: $10 = 2 \times 5$ , so 2 is always present
Check: Any number ?0 contains factor 10, which always includes prime 2 ✓

Common Mistakes

Avoid these frequent errors

Assuming other primes like 5 or 11 always appear
Don't assume that because 10 = 2 × 5, the prime 5 always appears in every ?0 number = wrong conclusion! While 10, 20, 30 contain 5, numbers like 40 = 2³ × 5 don't guarantee 5 in partial factorizations. Always remember only 2 is guaranteed since every ?0 number is even.

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 always a factor of numbers ending in 0?

Any number ending in 0 is automatically even (divisible by 2). Since 2 is prime, it will always appear in the prime factorization of these numbers.

What about 5? Doesn't 10 = 2 × 5?

While it's true that $10 = 2 \times 5$ , the question asks about factors that surely appear. Some ?0 numbers might have 5 grouped differently in their complete factorization, but 2 is always guaranteed.

Could 6 be a prime factor?

No! The number 6 is not prime because $6 = 2 \times 3$ . Only numbers with exactly two factors (1 and themselves) are prime.

How do I identify prime factors quickly?

Start with the smallest prime numbers: 2, 3, 5, 7, 11... Check if your number is divisible by each one. For ?0 numbers, you can immediately write down 2 as a factor!

What if the number is 10 itself?

Perfect example! $10 = 2 \times 5$ , so both 2 and 5 are prime factors. But across all possible ?0 numbers (10, 20, 30, 40...), only 2 appears in every single one.

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