Number Classification: Is 29 Prime or Composite?

Q: Why do I only need to check up to the square root?

If a number has a factor larger than its square root, it must also have a corresponding factor smaller than the square root. So checking up to $ \sqrt{29} \approx 5.4 $ covers all possibilities!

Q: What's the difference between prime and composite again?

Prime: Exactly 2 divisors (1 and itself)Composite: More than 2 divisorsSpecial case: 1 is neither prime nor composite

Q: How do I quickly check divisibility rules?

Use these shortcuts:Divisible by 2: Last digit is evenDivisible by 3: Sum of digits divisible by 3Divisible by 5: Ends in 0 or 5

Q: What if I can't find any factors - does that prove it's prime?

Yes! If you've tested all prime numbers up to the square root and found no factors, then your number is definitely prime. That's exactly what we did with 29!

Prime Number Testing with Square Root Method

Is the number equal to $n$ prime or composite?

$n=29$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Is the number prime or composite?

00:03 A prime number is only divisible by itself and 1

00:06 Therefore, if the number is divisible by another factor, it's not prime

00:10 The number has no other factors, meaning it's prime

00:14 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

Is the number equal to $n$ prime or composite?

$n=29$

Step-by-step solution

To determine whether the number $n = 29$ is prime or composite, we will check if $n$ has any divisors other than 1 and itself.

The definition of a prime number is one that has exactly two distinct positive divisors: 1 and itself. Conversely, a composite number has more than two distinct positive divisors.

First, observe that the square root of 29 is approximately 5.385. This tells us that we only need to check divisibility by all prime numbers less than or equal to 5. These primes are 2, 3, and 5.

Check divisibility by 2: 29 is odd, so it is not divisible by 2.
Check divisibility by 3: The sum of the digits of 29 is $2 + 9 = 11$ , which is not divisible by 3, so 29 is not divisible by 3.
Check divisibility by 5: 29 does not end in 0 or 5, so it is not divisible by 5.

Since 29 is not divisible by any of these primes, it has no divisors other than 1 and 29 itself. Therefore, 29 is a prime number.

Hence, the solution to the problem is that the number $n = 29$ is Prime.

Final Answer

Prime

Key Points to Remember

Essential concepts to master this topic

Definition: Prime numbers have exactly two divisors: 1 and themselves
Technique: Check divisibility by primes up to $\sqrt{29} \approx 5.4$
Check: Test 2, 3, and 5: none divide 29 evenly ✓

Common Mistakes

Avoid these frequent errors

Testing all numbers up to n-1
Don't check every number from 2 to 28 to see if it divides 29 = wasted time and effort! You only need to check up to the square root because factors come in pairs. Always test divisibility only up to $\sqrt{n}$ to save time.

Practice Quiz

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Which of the numbers is a prime number?

FAQ

Everything you need to know about this question

Why do I only need to check up to the square root?

If a number has a factor larger than its square root, it must also have a corresponding factor smaller than the square root. So checking up to $\sqrt{29} \approx 5.4$ covers all possibilities!

What if the number I'm testing is really large?

The square root method still works! For example, to test if 121 is prime, you only check divisors up to $\sqrt{121} = 11$ . You'll find that 11 divides 121, so it's composite.

Do I need to test all numbers or just prime numbers?

Only test prime numbers! If a composite number like 4 doesn't divide your number, then its factors (like 2) won't divide it either. Test: 2, 3, 5, 7, 11, 13...

What's the difference between prime and composite again?

Prime: Exactly 2 divisors (1 and itself)
Composite: More than 2 divisors
Special case: 1 is neither prime nor composite

How do I quickly check divisibility rules?

Use these shortcuts:

Divisible by 2: Last digit is even
Divisible by 3: Sum of digits divisible by 3
Divisible by 5: Ends in 0 or 5

What if I can't find any factors - does that prove it's prime?

Yes! If you've tested all prime numbers up to the square root and found no factors, then your number is definitely prime. That's exactly what we did with 29!

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