Question

Which of these numbers can be expressed as a product of two or more prime numbers? i) 15 ii) 34568 iii) (15 × 13) ​

Answer 1

Answer:

Required numbers are 15 and (15×13)

Step-by-step explanation:

We know, prime numbers mean those numbers which are divisible by 1 and by own only.

Here given three numbers are 15,34568 and (15×13).

For first number,

$15 = 5 \times 3$

Here 5 and 3 both are prime numbers.

So, we can express 15 as product of two or more prime numbers.

For second number,

34568 is divisible by 4 and 4 is not a prime number.

So,34568 can't express as product of two or more prime numbers.

For third numbers,

$15 \times 13 = 5 \times 3 \times 13$

Here 5,3 and 13 are prime numbers.

So,we can express (15×13) as product of two or more prime numbers.

Therefore, required numbers are 15 and (15×13)

Know more about prime numbers:

A prime (or prime number) is a natural number greater than 1 that is not the product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways to write it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is a sum because it is a product (2 × 2) in which both numbers are less than 4. Prime numbers are central to number theory because the Fundamental Theorem of Arithmetic: Every natural number greater than 1 is a prime number. itself, or it can be thought of as a product of primes that is unique in their order.

The quality of being first is called primacy. A simple but slow method to check the primality of a given number

n, called the sampling distribution, tests whether

n is a multiple of any integer between 2 and faster algorithms include the Miller-Rabin primality test, which is fast but has a low error rate, and the AKS primality test, which always gives the correct answer in polynomial time, but is too slow to be practical. Especially fast methods are available for numbers of special forms, such as Mersenne numbers. As of December 2018, the largest known prime number is the Mersenne prime number with 24,862,048 decimal places.

There are infinitely many primes, as Euclid showed around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of prime numbers in large natural numbers can be modeled statistically. The first result in this direction is the prime number theorem, proved at the end of the 19th century, which states that the probability that a randomly chosen large number is prime is inversely proportional to its number of digits, i.e. its logarithm.

Several historical questions related to prime numbers are still unresolved. These include Goldbach's conjecture, which states that every even number greater than 2 can be expressed as the sum of two primes, and the Evenness Conjecture, which states that there are infinitely many pairs of prime numbers with only one even number between them. Such questions gave impetus to the development of various branches of number theory, focusing on the analytical or algebraic aspects of numbers. Prime numbers are used in several routines in information technology, such as public key cryptography, which depends on the difficulty of factoring large numbers. In abstract algebra, objects that generally behave as prime numbers include prime elements and prime ideals.

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