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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.

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Last updated date: 22nd Jul 2024
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Answer
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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 $ .