Consecutive Odd Numbers with a Product of 323: A Step-by-Step Guide

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In this article, we will explore the problem of identifying two consecutive odd numbers that, when multiplied together, result in 323. This is a fascinating problem that can be approached through both algebraic and numerical methods. Let's break it down step-by-step.

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Approach through Numerical Estimation

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First, let's estimate the numbers using simple multiplication:

" "" "10×10 100" "15×15 225" "20 × 20 400" "" "

From these calculations, we can deduce that the product of two consecutive odd numbers is just below 20 but higher than 15. The only pair of consecutive odd numbers in this range is 17 and 19. Let's verify:

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19 × 17 323

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This is a quick solution, but let's explore the algebraic method as well.

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Algebraic Method

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Let's denote the two consecutive odd numbers as x and x 2. The product of these numbers is given by:

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x(x 2) 323

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This can be rewritten as:

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x^2 2x - 323 0

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Solving the Quadratic Equation

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To solve the quadratic equation, we will factorize it:

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x^2 19x - 17x - 323 0

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Grouping the terms, we get:

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x^2 19x - 17x - 323 0

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This simplifies to:

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x(x 19) - 17(x 19) 0

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Factoring out the common factor:

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(x 19)(x - 17) 0

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This gives us two solutions for x:

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x 19 0 Rightarrow x -19

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x - 17 0 Rightarrow x 17

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The relevant solution in the context of our problem (positive integers) is:

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x 17

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Therefore, the two consecutive odd numbers are:

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17 and 19

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Verification and Additional Methods

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To verify, we can also use a different approach:

" "" "Assume the first number as 2n-1 and the next consecutive odd number as 2n 1." "Calculate their product:" "(2n-1)(2n 1) 4n^2 - 1 323" "4n^2 324" "n^2 81" "n 9 (ignoring the negative value)" "So, the numbers are 2n-1 17 and 2n 1 19" "" "

Thus, the two consecutive odd numbers are conclusively identified as 17 and 19.

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Conclusion

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The problem of finding two consecutive odd numbers whose product is 323 can be effectively solved through both numerical estimation and algebraic methods. The numbers are 17 and 19. Proving the solution involves simple algebraic manipulations or straightforward multiplication checks. Understanding these methods can be valuable for solving similar types of number theory problems.