Find the smallest number that is divisible by first four positive composite numbers
|
I'm pretty sure the following statement is true: Show
But I'm having trouble proving it. Here is an attempt by induction. The property is true when $n=6$, and assume it's true for $n$. If $n+1$ isn't prime, the induction step is trivial, for the smallest prime bigger than $n+1$ is equal to the smallest prime bigger than $n$; call this prime $p$. But by hypothesis all composites smaller than $2p$ divide $n!$, hence $(n+1)!$. The harder case is when $n+1$ is prime. Let $q$ denote the next prime, i.e. the smallest prime bigger than $n+1$. We know by hypothesis that all composites smaller than $2(n+1)$ divide $n!$, hence $(n+1)!$. We also know $2(n+1)$ divides $(n+1)!$. To finish, we need to show that all composites $m$ strictly between $2(n+1)$ and $2q$ divide $(n+1)!$. This is where I get stuck. It certainly helps that the ratio $\frac{q}{n+1}$ can't be larger than 2 (by Bertrand's postulate; I imagine the bound can be sharpened but I know embarrassingly little number theory). It's also obvious that the prime factors of any such composite $m$ are all smaller than $q$. What I don't quite see is an argument to ensure the powers of those prime factors aren't too large. Feel free to give alternative approaches, rather than by induction, if there is a much simpler proof I've overlooked.
Jump to
NCERT Solutions Class 6 Mathematics Solutions for Exercise 3.5 in Chapter 3 - Playing With NumbersQ12) I am the smallest number, having four different prime factors. Can you find me? Answer: SOLUTION: The smallest four prime numbers are 2,3,5 and 7. hence, the required number is 2\times3\times5\times7=210
Video transcript "Hello guys, welcome to Lido homework today we'll do question number 12. Now. This is when I questioned or five immigration. So the question is I am the smallest prime number having four different prime factors. Can you find me? So as the numbers for different prime factors? So take the for smallest conjugate of prime numbers. So the for smallest conjugate of prime numbers are 2 3 5 and 7. So these are the prime factors of that of this number that we've defined. So as these are prime factors when you multiply all of these As you're going to get that number, so your answer is 2 into 3. Into 5 into 7 Therefore, your answer is 210. Thank you guys for watching. This was very easy question. If you have any doubts, please let me know in the comments below do not hesitate. Also, please like the video and subscribe the channel. Thank you so much. " Was This helpful?
 In math, composite numbers can be defined as numbers that have more than two factors. Numbers that are not prime are composite numbers because they are divisible by more than two numbers. Examples: Since 4 has more
than two factors. So, 4 is a composite number. Since 6 also has more than two factors. So, 6 is also a composite number. The divisibility test is a standard method used to find a composite number. In this test, the given number is divided by a smaller prime or composite number. If it is entirely divisible, the number is a composite number. For example, 48 = $2 \times 2 \times 2 \times 2 \times 3$ Since 48 is divisible by 2 and 3, hence it is a composite number. There are two types of composite numbers: Composite numbers with an odd digit in the unit’s place are odd composite numbers. In simple words, all the odd numbers that are not prime numbers are odd composite numbers. For example: 9, 15, 21, etc. Composite numbers with an even digit in the unit’s place are even composite numbers. In simple words, all the even numbers except 2 are even composite numbers. This is because no even number (except 2) can ever be a prime number. For example: 8, 12, 14, etc. 1. Check if 104 is a composite number or not. The given number 104 is divisible by 2 and hence have more than 2 factors. Therefore,
it is a composite number. 2. Check if 111 is a composite number. The given number is divisible by 3 and hence have more than 2 factors. Therefore, it is a composite number. 3. Check if 179 and 144 are composite numbers. Among the given numbers, 179 is not divisible by any number other than 1 and 179; therefore, it is not a composite number. 144 is divisible by 2, so it is a composite number. Practice Problems9 11 14 2 Correct answer is: 9 15 21 25 23 Correct answer is: 23 11 13 17 20 Correct answer is:
20 47 33 35 39 Correct answer is: 39 Conclusion Frequently Asked QuestionsCan a number be both prime and composite? No, a number cannot be both prime and composite. A prime number has exactly two factors, 1 and itself, while a composite number has more than two factors. So, all natural numbers (except 1) are either prime or composite, but not both. Any number multiplied by zero gives the product 0. Hence 0 has an infinite number of factors. To be composite, a number must have more than two factors but not an infinite number of them. Hence, 0 does not qualify as a composite number. What is the smallest composite number? 4 is the smallest composite number. Are all even numbers composite numbers? No. All even numbers except 2 are composite numbers. 2 has only two factors: 1 and 2. What are the first 4 composite numbers?The first few composite numbers (sometimes called "composites" for short) are 4, 6, 8, 9, 10, 12, 14, 15, 16, ... (OEIS A002808), whose prime decompositions are summarized in the following table. Note that the number 1 is a special case which is considered to be neither composite nor prime.
What is the 4 smallest composite numbers?Solution: The first 5 composite numbers are 4, 6, 8, 9, 10.
What is the smallest positive composite number?4 is the smallest composite number. Each composite number can be written as a product of two or more primes.. Composite numbers are divisible by other composite numbers. All composite numbers are always divisible by 1 and the number itself.
What is the smallest composite number 1 2 3 4 answer?The smallest composite number is 4. This is because the factors of 4 are 1,2, and 4 . Since it has at least three factors, it is the smallest composite number. The number 1 is neither prime nor composite.
|
