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

Last updated date: 22nd Jul 2024
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Hint: First we have to define what the terms we need to solve the problem are.
We will write the prime factors of the number $1729$ and then arrange them in ascending order,
Now further check the difference between two consecutive prime numbers to find the relation between them.

Complete step by step solution:
Write the prime factorization of $1729$ and arrange them in ascending order.
We will start with $1729$ , to write prime factorization of a number first divided by $2$ and continue until the remainder does not zero so we will get a remainder because $1729$ is an odd number.
Then start dividing by $3$ from same process and continue with the series or prime numbers $5,7,9,11$
Until only numbers are left is prime numbers and taking product of all prime numbers.(fact to count each prime number)
Thus, we get the prime factorization of $1729$ is $1729$ $= 7 \times 13 \times 19$
Hence the prime factors are $1729$ are $7,13$ and $19$ and these are in ascending order too
Now we need to subtract the two consecutive prime numbers.
So, we have $13 - 7 = 6$ and $19 - 13 = 6$
Hence, we observe the difference between two consecutive primes is equal and constant.
Thus, the prime factors of $1729$ arranged in ascending order are $7,13$ and $19$ and the difference between two consecutive factors is constant or equal to $6$

Note: Only if we subtract the ascending will lead to equal or constant, if we do the same things in a descending order, it may be different (not equal or constant) and prime number is a number which is multiplied only by $1$ .