Pinpointing Prime Factors: Which Always Appears in a Two-Digit Number?

Prime Factorization with Two-Digit Numbers

I am a two-digit number ?0 ?0

Which prime factor will surely appear among my first factors?

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Step-by-step video solution

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00:00 Which factor definitely appears in the primary factors of the number?
00:04 The ones digit is 0
00:06 Let's try dividing 10 by each of the factors and see what's possible
00:20 Let's try dividing various tens by factor 5 and see what divides
00:25 In fact, any number with tens digit and ones digit being 0 is divisible by 5
00:30 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

I am a two-digit number ?0 ?0

Which prime factor will surely appear among my first factors?

2

Step-by-step solution

To solve this problem, follow these steps:

  • Step 1: Recognize that a two-digit number ending in 0 can be expressed as 10×n 10 \times n , where n n is an integer from 1 to 9.
  • Step 2: Since the number ends in 0, it is essentially divisible by 10.
  • Step 3: The prime factorization of 10 is 10=2×5 10 = 2 \times 5 .
  • Step 4: Among these prime factors, we are asked which one surely appears. That is, any number ending in 0 will certainly include 5 as a prime factor.
  • Step 5: Therefore, the prime factor that will surely appear is 5 5 .

Therefore, the solution to the problem is that the prime factor is 5 5 .

3

Final Answer

5 5

Key Points to Remember

Essential concepts to master this topic
  • Pattern: Numbers ending in 0 always contain factors 2 and 5
  • Method: Any number n0 n0 equals 10×n 10 \times n , so includes 2×5 2 \times 5
  • Verify: Check that 20, 30, 40 all divide by 5 evenly ✓

Common Mistakes

Avoid these frequent errors
  • Thinking only the tens digit matters for prime factors
    Don't ignore the zero and only look at the tens digit = missing guaranteed factors! The zero makes the number divisible by 10, which always contains prime factors 2 and 5. Always recognize that any number ending in 0 must include both 2 and 5 as prime 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 is 5 always a factor but not 2?

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Actually, both 2 and 5 are always factors of numbers ending in 0! Since any two-digit number ending in 0 is divisible by 10, and 10=2×5 10 = 2 \times 5 , both primes must appear.

What about the other answer choices like 3, 7, and 11?

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These primes might appear in some two-digit numbers ending in 0, but they're not guaranteed. For example, 20 doesn't contain 3, 7, or 11 as factors, but it does contain 2 and 5.

How can I quickly identify all prime factors of a number ending in 0?

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Start with the guaranteed factors 2 and 5, then check if the tens digit contributes additional prime factors. For example: 30 = 2 × 3 × 5, so it has prime factors 2, 3, and 5.

Does this rule work for larger numbers ending in 0?

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Yes! Any number ending in 0 (whether 2-digit, 3-digit, or larger) is divisible by 10, so it will always contain prime factors 2 and 5.

What if the question asked which prime factor might NOT appear?

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Then you'd look for primes that don't always divide numbers ending in 0. Primes like 3, 7, and 11 might not appear in every case, making them possible answers for that type of question.

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