The Product of Two Consecutive Odd Integers Is 143 – Find the Integers

Have you ever wondered how to solve a simple yet intriguing math puzzle? One classic example is finding two consecutive odd integers whose product equals 143. In this article, we’ll explore how to identify these integers step by step and understand the logic behind their connection to the number 143.

Understanding the Context

Understanding the Problem

We are told that the product of two consecutive odd integers is 143. Let’s define these integers algebraically:

Let the first odd integer be
x
Then the next consecutive odd integer is x + 2 (since odd numbers are two units apart).

Thus, we write the equation:
x × (x + 2) = 143

Solved The product of two consecutive odd integers is 143 . | Chegg.com

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Solved The product of two consecutive odd integers is 143 . | Chegg.com
Product of Two Consecutive Integers Calculator (Natural, Odd & Even ...
Solved: Two positive, consecutive, odd integers have a product of 143 ...
The product of two consecutive odd integers is 99 . | Chegg.com
Solved Find two consecutive positive odd integers whose | Chegg.com

Key Insights

Setting Up the Equation

Expand the equation:
x² + 2x = 143

Bring all terms to one side to form a quadratic equation:
x² + 2x – 143 = 0

Continue Reading

Question:** A mathematician working on algebraic topology defines a function \( f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 \). Determine the remainder when \( f(x) \) is divided by \( x - 1 \).
Solution:** To find the remainder when \( f(x) \) is divided by \( x - 1 \), we use the Remainder Theorem, which states that the remainder of the division of a polynomial \( f(x) \) by \( x - c \) is \( f(c) \). Here, \( c = 1 \), so we calculate \( f(1) \):
f(1) = 1^4 - 4 \cdot 1^3 + 6 \cdot 1^2 - 4 \cdot 1 + 1 = 1 - 4 + 6 - 4 + 1 = 0

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Final Thoughts

Solving the Quadratic Equation

We can solve this using factoring, completing the square, or the quadratic formula. Let's attempt factoring.

We need two numbers that:

  • Multiply to –143
  • Add to 2 (the coefficient of x)

Factoring 143:
143 = 11 × 13
So, –11 and +13 multiply to –143 and add to 2 ✅

Thus, factor the equation:
(x + 11)(x – 13) = 0

Wait — actually, (x + 11)(x – 13) = x² – 2x – 143 — not our equation. We need (x + 11)(x – 13) = x² – 2x – 143, but our equation is x² + 2x – 143.

Let’s correct: we want two numbers that multiply to –143 and add to +2. Try:

11 and –13? → no, add to –2
Try –11 and 13? → add to 2 → yes! But signs differ.

Actually, correct factoring candidates: