Question

Find all the prime factors of 1729 and arrange them in ascending order. Now state the relation, if any; between two consecutive prime factors.

Answer 1

Hint: Write the prime factorization of the number 1729 and then arrange them in ascending order. Check the difference between two consecutive prime numbers to find the relation or find any other relation between them.

Complete step-by-step answer:
We have to write all prime factors of 1729 and arrange them in ascending order.

We will begin by writing the prime factorization of number 1729.

To write the prime factorization of any number, start by dividing the number by the first prime number, which is 2 and then continue to divide by 2 until you get a number which is not divisible by 2 (which means that you get a decimal or remainder on dividing the number by 2). Then start dividing the number by the next prime number which is 3. Continue dividing the number by \[3\] until you get a number which is not divisible by 3. Thus, continuing the process, keep dividing the numbers by series of prime numbers \[5,7,...\]until the only numbers left are prime numbers. Write the given number as a product of all the prime numbers (considering the fact to count each prime number as many times as it divides the given number) to get the prime factorization of the given number.

Thus, the prime factorization of 1729 is \[1729=7\times 13\times 19\].

So, the prime factors of 1729 are 7, 13 and 19. We observe that they are already arranged in ascending order.

We will now find the relation between two consecutive prime numbers.

We will subtract the two consecutive prime numbers, such that the smaller number is subtracted from the larger one

So, we have \[13-7=6,19-13=6\].

Thus, we observe that the difference between two consecutive prime numbers is a constant.

Hence, the prime factors of 1729, arranged in ascending order are 7, 13, 19 and the difference between two consecutive factors is a constant, 6.

Note: We can look for other relations between two consecutive prime numbers as well, such as, and two consecutive prime numbers are co prime. It’s necessary to arrange the prime numbers in ascending order and then find a relation between them.