We start by saying ,$$$$leaving 2 all prime numbers have odd units place , $$$$possible combinations for 3 consecutive prime numbers will be. $$n1,n3,n5$$ $$n3,n5,n7$$ $$n5,n7,n9$$ $$n7,n9,(n+1)1$$ $$n9,(n+1)1,(n+1)3$$ As all the numbers $\gt 5$ with units place 5 are divisible by 5, so by eliminating cases with n5 we are left with, $$n7,n9,(n+1)1$$ $$n9,(n+1)1,(n+1)3$$
So on our possible cases if $n=3k$
For $n=3k +1$
For $n=3k+2$
We all cases are disapproved we thus say there are no three consecutive prime numbers $\gt 5$.
Question 2 Exercise 3.5 Next
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 number theory, a sphenic number (from Ancient Greek: σφήνα, 'wedge') is a positive integer that is the product of three distinct prime numbers. Because of there are infinitely many prime numbers, there are also infinitely many sphenic numbers.
A sphenic number is a product pqr where p, q, and r are three distinct prime numbers. In other words, the sphenic numbers are the square-free 3-almost primes.
The smallest sphenic number is 30 = 2 × 3 × 5, the product of the smallest three primes.
The first few sphenic numbers are
As of October 2020[ref] the largest known sphenic number is
It is the product of the three largest known primes.
All sphenic numbers have exactly eight divisors. If we express the sphenic number as
n
=
p
⋅
q
⋅
r
{\displaystyle n=p\cdot q\cdot r}
, where p, q, and r are distinct primes, then the set of divisors of n will be:
The converse does not hold. For example, 24 is not a sphenic number, but it has exactly eight divisors.
All sphenic numbers are by definition squarefree, because the prime factors must be distinct.
The Möbius function of any sphenic number is −1.
The cyclotomic polynomials
Φ
n
(
x
)
{\displaystyle \Phi _{n}(x)}
, taken over all sphenic numbers n, may contain arbitrarily large coefficients[1] (for n a product of two primes the coefficients are
±
1
{\displaystyle \pm 1}
or 0).
Any multiple of a sphenic number (except by 1) isn't a sphenic number. This is easily provable by the multiplication process adding another prime factor, or squaring an existing factor.
The first case of two consecutive sphenic integers is 230 = 2×5×23 and 231 = 3×7×11. The first case of three is 1309 = 7×11×17, 1310 = 2×5×131, and 1311 = 3×19×23. There is no case of more than three, because every fourth consecutive positive integer is divisible by 4 = 2×2 and therefore not squarefree.
The numbers 2013 (3×11×61), 2014 (2×19×53), and 2015 (5×13×31) are all sphenic. The next three consecutive sphenic years will be 2665 (5×13×41), 2666 (2×31×43) and 2667 (3×7×127) (sequence A165936 in the OEIS).
Building numbers from smaller building blocks: Any counting number, other than 1, can be built by adding two or more smaller counting numbers. But only some counting numbers can be composed by multiplying two or more smaller counting numbers. Prime and composite numbers: We can build 36 from 9 and 4 by multiplying; or we can build it from 6 and 6; or from 18 and 2; or even by multiplying 2 × 2 × 3 × 3. Numbers like 10 and 36 and 49 that can be composed as products of smaller counting numbers are called composite numbers. Some numbers can’t be built from smaller pieces this way. For example, he only way to build 7 by multiplying and by using only counting numbers is 7 × 1. To “build” 7, we must use 7! So we’re not really composing it from smaller building blocks; we need it to start with. Numbers like this are called prime numbers. Informally, primes are numbers that can’t be made by multiplying other numbers. That captures the idea well, but is not a good enough definition, because it has too many loopholes. The number 7 can be composed as the product of other numbers: for example, it is 2 × 3. To capture the idea that “7 is not divisible by 2,” we must make it clear that we are restricting the numbers to include only the counting numbers: 1, 2, 3…. A formal definitionA prime number is a positive integer that has exactly two distinct whole number factors (or divisors), namely 1 and the number itself. Clarifying two common confusionsTwo common confusions:
The number 1 is not prime. Why not?Well, the definition rules it out. It says “two distinct whole-number factors” and the only way to write 1 as a product of whole numbers is 1 × 1, in which the factors are the same as each other, that is, not distinct. Even the informal idea rules it out: it cannot be built by multiplying other (whole) numbers. But why rule it out?! Students sometimes argue that 1 “behaves” like all the other primes: it cannot be “broken apart.” And part of the informal notion of prime — we cannot compose 1 except by using it, so it must be a building block — seems to make it prime. Why not include it? Mathematics is not arbitrary. To understand why it is useful to exclude 1, consider the question “How many different ways can 12 be written as a product using only prime numbers?” Here are several ways to write 12 as a product but they don’t restrict themselves to prime numbers. 3 × 4 4 × 3 1 × 12 1 × 1 x 12 2 × 6 1 × 1 × 1 × 2 × 6 Using 4, 6, and 12 clearly violates the restriction to be “using only prime numbers.” But what about these? 3 × 2 × 2 2 × 3 × 2 1 × 2 × 3 × 2 2 × 2 × 3 × 1 × 1 × 1 × 1 Well, if we include 1, there are infinitely many ways to write 12 as a product of primes. In fact, if we call 1 a prime, then there are infinitely many ways to write any number as a product of primes. Including 1 trivializes the question. Excluding it leaves only these cases: 3 × 2 × 2 2 × 3 × 2 2 × 2 × 3 This is a much more useful result than having every number be expressible as a product of primes in an infinite number of ways, so we define prime in such a way that it excludes 1. The number 2 is prime. Why? Students sometimes believe that all prime numbers are odd. If one works from “patterns” alone, this is an easy slip to make, as 2 is the only exception, the only even prime. One proof: Because 2 is a divisor of every even number, every even number larger than 2 has at least three distinct positive divisors. Another common question: “All even numbers are divisible by 2 and so they’re not prime; 2 is even, so how can it be prime?” Every whole number is divisible by itself and by 1; they are all divisible by something. But if a number is divisible only by itself and by 1, then it is prime. So, because all the other even numbers are divisible by themselves, by 1, and by 2, they are all composite (just as all the positive multiples of 3, except 3, itself, are composite). Mathematical backgroundUnique prime factorization and factor treesThe question “How many different ways can a number be written as a product using only primes?” (see why 1 is not prime) becomes even more interesting if we ask ourselves whether 3 × 2 × 2 and 2 × 2 × 3 are different enough to consider them “different ways.” If we consider only the set of numbers used — in other words, if we ignore how those numbers are arranged — we come up with a remarkable, and very useful fact (provable). Every whole number greater than 1 can be factored into a unique set of primes. There is only one set of prime factors for any whole number.Primes and rectanglesIt is possible to arrange 12 square tiles into three distinct rectangles. Seven square tiles can be arranged in many ways, but only one arrangement makes a rectangle. How many primes are there?From 1 through 10, there are 4 primes: 2, 3, 5, and 7. From 11 through 20, there are again 4 primes: 11, 13, 17, and 19. From 21 through 30, there are only 2 primes: 23 and 29. From 31 through 40, there are again only 2 primes: 31 and 37. From 91 through 100, there is only one prime: 97. It looks like they’re thinning out. That even seems to make sense; as numbers get bigger, there are more little building blocks from which they might be made. Do the primes ever stop? Suppose for a moment that they do eventually stop. In other words, suppose that there were a “greatest prime number” — let’s call it p. Well, if we were to multiply together all of the prime numbers we already know (all of them from 2 to p), and then add 1 to that product, we would get a new number — let’s call it q — that is not divisible by any of the prime numbers we already know about. (Dividing by any of those primes would result in a remainder of 1.) So, either q is prime itself (and certainly greater than p) or it is divisible by some prime we have not yet listed (which, therefore, must also be greater than p). Either way, the assumption that there is a greatest prime — p was supposedly our greatest prime number — leads to a contradiction! So that assumption must be wrong there is no “greatest prime number”; the primes never stop. Suppose we imagine that 11 is the largest prime. 2 × 3 × 5 × 7 × 11 + 1 = 2311 —- Prime! No number (except 1) divides 2311 with zero remainder, so 11 is not the largest prime.Suppose we imagine that 13 is the largest prime. 2 × 3 × 5 × 7 × 11 × 13 + 1 = 30031 —- Not prime! But 59 × 509 = 30031, and both 59 and 509 are prime, and both are greater than 13, so 13 is not the largest prime. |
What is the smallest positive odd integer that is the product of three different prime numbers
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