From 0b49549ef903ba7c5f7ae7376a87aa18221b90b4 Mon Sep 17 00:00:00 2001 From: Claude Date: Mon, 13 Jul 2026 09:08:11 +0000 Subject: [PATCH] Add Quant number-properties: odds/evens/signs lesson + 5 questions Adds the odds-evens-signs subtopic (previously empty in the taxonomy): one concept lesson covering parity of sums/products, consecutive-integer parity, and sign rules for products/quotients, plus five original problem-solving questions (easy->hard) with worked explanations and hints. Rebuilds the content index. Co-Authored-By: Claude Opus 4.8 Claude-Session: https://claude.ai/code/session_01SqS4XiZnt8Ar6y1FM4c1uW --- ...uant-number-properties-odds-evens-signs.md | 78 ++++++++ ...number-properties-odds-evens-signs-0017.md | 45 +++++ ...number-properties-odds-evens-signs-0018.md | 45 +++++ ...number-properties-odds-evens-signs-0019.md | 43 +++++ ...number-properties-odds-evens-signs-0020.md | 46 +++++ ...number-properties-odds-evens-signs-0021.md | 44 +++++ docs/data/content.json | 175 +++++++++++++++++- 7 files changed, 472 insertions(+), 4 deletions(-) create mode 100644 content/lessons/quant/quant-number-properties-odds-evens-signs.md create mode 100644 content/questions/quant/quant-number-properties-odds-evens-signs-0017.md create mode 100644 content/questions/quant/quant-number-properties-odds-evens-signs-0018.md create mode 100644 content/questions/quant/quant-number-properties-odds-evens-signs-0019.md create mode 100644 content/questions/quant/quant-number-properties-odds-evens-signs-0020.md create mode 100644 content/questions/quant/quant-number-properties-odds-evens-signs-0021.md diff --git a/content/lessons/quant/quant-number-properties-odds-evens-signs.md b/content/lessons/quant/quant-number-properties-odds-evens-signs.md new file mode 100644 index 0000000..b70abbc --- /dev/null +++ b/content/lessons/quant/quant-number-properties-odds-evens-signs.md @@ -0,0 +1,78 @@ +--- +id: quant-number-properties-odds-evens-signs +section: quant +topic: number-properties +subtopic: odds-evens-signs +title: "Odds, Evens & Signs" +tags: [parity, odd-even, signs, number-properties] +author: openmat +reviewers: [] +status: in-review +original: true +license: CC-BY-SA-4.0 +--- + +## Overview + +Many GMAT Focus Quant questions never ask you to compute a value — they ask what **must** be +true about whether a number is odd or even, or positive or negative. These **parity** and **sign** +questions look abstract, but they collapse to a handful of fixed rules. Once you know the rules you +can answer in seconds, and when you don't, you can always fall back on **plugging in small numbers**. + +## Core concepts + +**Parity of sums and differences.** Addition and subtraction have the *same* parity behavior: + +\[\text{even} \pm \text{even} = \text{even}, \quad \text{odd} \pm \text{odd} = \text{even}, \quad \text{even} \pm \text{odd} = \text{odd}\] + +The takeaway: \(a + b\) and \(a - b\) always have the **same** parity. A sum (or difference) is odd +**only** when exactly one of the two numbers is odd. + +**Parity of products.** A product is even if **any** factor is even; it is odd **only** when **every** +factor is odd: + +\[\text{even} \times \text{anything} = \text{even}, \qquad \text{odd} \times \text{odd} = \text{odd}\] + +**Two consecutive integers.** \(k\) and \(k+1\) are always one odd and one even, so their product +\(k(k+1)\) is **always even**. This is why expressions like \(k^2 + k = k(k+1)\) are guaranteed even. + +**Signs of products and quotients.** For multiplication and division, only the *count* of negative +factors matters: + +- An **even** number of negative factors → **positive** result. +- An **odd** number of negative factors → **negative** result. + +So a product of three numbers is negative exactly when one or three of them are negative — never +when exactly two are. + +**Zero is even, and it has no sign.** \(0\) is an even integer. It is neither positive nor negative, +and any product containing a \(0\) is \(0\) — a case that quietly breaks "must be positive/negative" +claims, so watch for whether the problem says the numbers are *nonzero*. + +## Worked examples + +**Parity of an expression.** Is \(3p + 5q\) odd or even? The coefficients don't change parity: +\(3p\) has the parity of \(p\), and \(5q\) has the parity of \(q\). So \(3p + 5q\) has the parity of +\(p + q\) — it's even exactly when \(p\) and \(q\) share the same parity. + +**Sign of a quotient.** If \(a < 0\), \(b > 0\), \(c < 0\), then \(\dfrac{ab}{c}\) has two negative +inputs (\(a\) and \(c\)) — an even count — so the result is **positive**. + +## Common traps + +- **Assuming subtraction behaves differently from addition.** \(x - y\) has the *same* parity as + \(x + y\). If \(x + y\) is odd, so is \(x - y\). +- **Forgetting the odd-count rule for signs.** "Exactly two negatives" gives a **positive** product, + not a negative one. +- **Ignoring zero.** If the problem doesn't say "nonzero," a variable could be \(0\) — which is even + and sign-less, and often kills a "must be positive" answer. +- **Testing only one case.** When you plug in numbers to eliminate a "must be true" choice, try both + parities and both signs; one clean counterexample is enough to reject a choice. + +## Key takeaways + +- \(a + b\) and \(a - b\) always share parity; a sum/difference is odd only when exactly one term is odd. +- A product is odd only if every factor is odd; a single even factor makes it even. +- Consecutive integers \(k(k+1)\) are always even. +- Sign of a product/quotient depends only on the parity of the *count* of negatives: even count → positive, odd count → negative. +- Zero is even and sign-less — check whether variables are allowed to be zero. diff --git a/content/questions/quant/quant-number-properties-odds-evens-signs-0017.md b/content/questions/quant/quant-number-properties-odds-evens-signs-0017.md new file mode 100644 index 0000000..69916f3 --- /dev/null +++ b/content/questions/quant/quant-number-properties-odds-evens-signs-0017.md @@ -0,0 +1,45 @@ +--- +id: quant-number-properties-odds-evens-signs-0017 +section: quant +topic: number-properties +subtopic: odds-evens-signs +type: problem-solving +difficulty: easy +tags: [parity, odd-even, must-be-true] +choices: + A: "\(x - y\)" + B: "\(xy\)" + C: "\(x^2 + y^2\)" + D: "\(x + 2y\)" + E: "\(2x + y\)" +answer: B +author: openmat +reviewers: [] +status: in-review +original: true +license: CC-BY-SA-4.0 +--- + +## Question + +If \(x\) and \(y\) are integers and \(x + y\) is odd, which of the following must be even? + +## Explanation + +If \(x + y\) is odd, then exactly one of \(x\) and \(y\) is odd and the other is even. + +- **A. \(x - y\):** subtraction has the same parity as addition, so \(x - y\) is **odd**, not even. + This is the classic trap. +- **B. \(xy\):** the product of an even number and an odd number is always **even**. This must be even. ✓ +- **C. \(x^2 + y^2\):** a square keeps the parity of its base, so \(x^2 + y^2\) has the same parity as + \(x + y\), which is **odd**. +- **D. \(x + 2y\):** \(2y\) is even, so this has the parity of \(x\). Since \(x\) could be the odd one + or the even one, this isn't guaranteed either way. +- **E. \(2x + y\):** similarly has the parity of \(y\), which isn't fixed. + +Only **B** is guaranteed even. + +## Hints + +- If \(x + y\) is odd, one of the two is even and the other is odd — write that down first. +- A product is even whenever *any* factor is even. diff --git a/content/questions/quant/quant-number-properties-odds-evens-signs-0018.md b/content/questions/quant/quant-number-properties-odds-evens-signs-0018.md new file mode 100644 index 0000000..135a40e --- /dev/null +++ b/content/questions/quant/quant-number-properties-odds-evens-signs-0018.md @@ -0,0 +1,45 @@ +--- +id: quant-number-properties-odds-evens-signs-0018 +section: quant +topic: number-properties +subtopic: odds-evens-signs +type: problem-solving +difficulty: easy +tags: [signs, must-be-true, inequalities] +choices: + A: "\(m > 0\) and \(n > 0\)" + B: "\(m < 0\) and \(n < 0\)" + C: "\(m\) and \(n\) have opposite signs" + D: "\(m = -n\)" + E: "exactly one of \(m, n\) is zero" +answer: B +author: openmat +reviewers: [] +status: in-review +original: true +license: CC-BY-SA-4.0 +--- + +## Question + +If \(mn > 0\) and \(m + n < 0\), which of the following must be true? + +## Explanation + +Because \(mn > 0\), the product is positive, so \(m\) and \(n\) have the **same** sign (and neither is +zero — a zero would make the product \(0\), not positive). + +Same sign leaves two possibilities: both positive or both negative. But \(m + n < 0\) is negative, +and two positive numbers can't add to something negative. So both must be **negative**. + +- **A** is the opposite conclusion. +- **C** would make the product negative, contradicting \(mn > 0\). +- **D** (\(m = -n\)) would give \(m + n = 0\), not less than \(0\). +- **E** would make \(mn = 0\), not positive. + +The answer is **B**. + +## Hints + +- A positive product means the two numbers share a sign. +- Can two positive numbers sum to a negative? Use that to pick between the two same-sign cases. diff --git a/content/questions/quant/quant-number-properties-odds-evens-signs-0019.md b/content/questions/quant/quant-number-properties-odds-evens-signs-0019.md new file mode 100644 index 0000000..9eb57dc --- /dev/null +++ b/content/questions/quant/quant-number-properties-odds-evens-signs-0019.md @@ -0,0 +1,43 @@ +--- +id: quant-number-properties-odds-evens-signs-0019 +section: quant +topic: number-properties +subtopic: odds-evens-signs +type: problem-solving +difficulty: medium +tags: [parity, consecutive-integers, must-be-true] +choices: + A: "\(k^2 + 1\)" + B: "\(k^2 + k\)" + C: "\(2k + 1\)" + D: "\(k^2 + k + 1\)" + E: "\(3k\)" +answer: B +author: openmat +reviewers: [] +status: in-review +original: true +license: CC-BY-SA-4.0 +--- + +## Question + +If \(k\) is a positive integer, which of the following is always even? + +## Explanation + +Test each choice against both parities of \(k\), or use parity rules directly. + +- **A. \(k^2 + 1\):** if \(k\) is even, \(k^2\) is even and \(k^2 + 1\) is **odd**. Not always even. +- **B. \(k^2 + k = k(k+1)\):** this is the product of two **consecutive** integers, one of which is + always even. So \(k(k+1)\) is **always even**. ✓ +- **C. \(2k + 1\):** \(2k\) is even, so \(2k + 1\) is always **odd**. +- **D. \(k^2 + k + 1 = k(k+1) + 1\):** an even number plus \(1\) is always **odd**. +- **E. \(3k\):** has the parity of \(k\); if \(k\) is odd (e.g. \(k = 1\)), \(3k = 3\) is **odd**. + +Only **B** is guaranteed even. + +## Hints + +- Try \(k = 1\) and \(k = 2\) in each choice and keep only the ones that stay even. +- Factor \(k^2 + k\) — what do you know about the product of two consecutive integers? diff --git a/content/questions/quant/quant-number-properties-odds-evens-signs-0020.md b/content/questions/quant/quant-number-properties-odds-evens-signs-0020.md new file mode 100644 index 0000000..77d8bb6 --- /dev/null +++ b/content/questions/quant/quant-number-properties-odds-evens-signs-0020.md @@ -0,0 +1,46 @@ +--- +id: quant-number-properties-odds-evens-signs-0020 +section: quant +topic: number-properties +subtopic: odds-evens-signs +type: problem-solving +difficulty: hard +tags: [signs, product, cannot-be-true] +choices: + A: "All three integers are negative." + B: "Exactly one of the integers is negative." + C: "Exactly two of the integers are negative." + D: "One of the integers equals \(-1\)." + E: "The sum of the three integers is positive." +answer: C +author: openmat +reviewers: [] +status: in-review +original: true +license: CC-BY-SA-4.0 +--- + +## Question + +The product of three integers is negative. Which of the following CANNOT be true? + +## Explanation + +A product is negative only when it is nonzero and the **count of negative factors is odd** — so with +three integers, either **one** or **all three** are negative. (None can be zero, or the product would +be \(0\).) + +- **A. All three negative:** three negatives is an odd count → negative product. **Possible.** +- **B. Exactly one negative:** one negative, two positive → negative product. **Possible.** +- **C. Exactly two negative:** two negatives is an *even* count, so those two multiply to a positive, + and the third (nonzero) factor keeps it positive. The product would be **positive**, never negative. + This **cannot** be true. ✓ +- **D. One equals \(-1\):** e.g. \(-1, 2, 3\) gives \(-6\). **Possible.** +- **E. Sum is positive:** e.g. \(-1, 5, 5\) gives product \(-25\) with sum \(9 > 0\). **Possible.** + +The answer is **C**. + +## Hints + +- A negative product needs an *odd* number of negative factors. +- Multiply out "exactly two negatives" with a small example — what sign do you always get? diff --git a/content/questions/quant/quant-number-properties-odds-evens-signs-0021.md b/content/questions/quant/quant-number-properties-odds-evens-signs-0021.md new file mode 100644 index 0000000..51a2062 --- /dev/null +++ b/content/questions/quant/quant-number-properties-odds-evens-signs-0021.md @@ -0,0 +1,44 @@ +--- +id: quant-number-properties-odds-evens-signs-0021 +section: quant +topic: number-properties +subtopic: odds-evens-signs +type: problem-solving +difficulty: medium +tags: [parity, coefficients, must-be-true] +choices: + A: "\(pq\)" + B: "\(p + q\)" + C: "\(2p + q\)" + D: "\(p + 1\)" + E: "\(pq + 1\)" +answer: B +author: openmat +reviewers: [] +status: in-review +original: true +license: CC-BY-SA-4.0 +--- + +## Question + +If \(p\) and \(q\) are integers and \(3p + 5q\) is even, which of the following must be even? + +## Explanation + +An odd coefficient doesn't change parity: \(3p\) has the parity of \(p\), and \(5q\) has the parity of +\(q\). So \(3p + 5q\) has the parity of \(p + q\). For that to be even, \(p\) and \(q\) must have the +**same parity** (both odd or both even). + +- **A. \(pq\):** both even → even, but both odd → odd. Not guaranteed. +- **B. \(p + q\):** same parity means their sum is **even** every time. This must be even. ✓ +- **C. \(2p + q\):** \(2p\) is even, so this has the parity of \(q\), which could be odd or even. +- **D. \(p + 1\):** flips the parity of \(p\); could be either. +- **E. \(pq + 1\):** since \(pq\) isn't fixed (see A), neither is \(pq + 1\). + +The answer is **B**. + +## Hints + +- Drop the coefficients: \(3p + 5q\) has the same parity as \(p + q\). +- If \(p + q\) is even, what does that tell you about whether \(p\) and \(q\) match in parity? diff --git a/docs/data/content.json b/docs/data/content.json index 9ef981c..eb6f289 100644 --- a/docs/data/content.json +++ b/docs/data/content.json @@ -7,8 +7,8 @@ "stats": { "quant": { "label": "Quantitative", - "questions": 16, - "lessons": 5 + "questions": 21, + "lessons": 6 }, "verbal": { "label": "Verbal", @@ -22,8 +22,8 @@ } }, "counts": { - "questions": 32, - "lessons": 8 + "questions": 37, + "lessons": 9 }, "questions": [ { @@ -418,6 +418,156 @@ "If \\(N = p^a q^b\\), the divisor count is \\((a+1)(b+1)\\). Remember the \"+1\" accounts for using the prime zero times." ] }, + { + "id": "quant-number-properties-odds-evens-signs-0017", + "section": "quant", + "topic": "number-properties", + "subtopic": "odds-evens-signs", + "type": "problem-solving", + "difficulty": "easy", + "tags": [ + "parity", + "odd-even", + "must-be-true" + ], + "choices": { + "A": "\\(x - y\\)", + "B": "\\(xy\\)", + "C": "\\(x^2 + y^2\\)", + "D": "\\(x + 2y\\)", + "E": "\\(2x + y\\)" + }, + "answer": "B", + "author": "openmat", + "reviewers": [], + "status": "in-review", + "prompt": "If \\(x\\) and \\(y\\) are integers and \\(x + y\\) is odd, which of the following must be even?", + "explanation": "If \\(x + y\\) is odd, then exactly one of \\(x\\) and \\(y\\) is odd and the other is even.\n\n- **A. \\(x - y\\):** subtraction has the same parity as addition, so \\(x - y\\) is **odd**, not even.\n This is the classic trap.\n- **B. \\(xy\\):** the product of an even number and an odd number is always **even**. This must be even. ✓\n- **C. \\(x^2 + y^2\\):** a square keeps the parity of its base, so \\(x^2 + y^2\\) has the same parity as\n \\(x + y\\), which is **odd**.\n- **D. \\(x + 2y\\):** \\(2y\\) is even, so this has the parity of \\(x\\). Since \\(x\\) could be the odd one\n or the even one, this isn't guaranteed either way.\n- **E. \\(2x + y\\):** similarly has the parity of \\(y\\), which isn't fixed.\n\nOnly **B** is guaranteed even.", + "hints": [ + "If \\(x + y\\) is odd, one of the two is even and the other is odd — write that down first.", + "A product is even whenever *any* factor is even." + ] + }, + { + "id": "quant-number-properties-odds-evens-signs-0018", + "section": "quant", + "topic": "number-properties", + "subtopic": "odds-evens-signs", + "type": "problem-solving", + "difficulty": "easy", + "tags": [ + "signs", + "must-be-true", + "inequalities" + ], + "choices": { + "A": "\\(m > 0\\) and \\(n > 0\\)", + "B": "\\(m < 0\\) and \\(n < 0\\)", + "C": "\\(m\\) and \\(n\\) have opposite signs", + "D": "\\(m = -n\\)", + "E": "exactly one of \\(m, n\\) is zero" + }, + "answer": "B", + "author": "openmat", + "reviewers": [], + "status": "in-review", + "prompt": "If \\(mn > 0\\) and \\(m + n < 0\\), which of the following must be true?", + "explanation": "Because \\(mn > 0\\), the product is positive, so \\(m\\) and \\(n\\) have the **same** sign (and neither is\nzero — a zero would make the product \\(0\\), not positive).\n\nSame sign leaves two possibilities: both positive or both negative. But \\(m + n < 0\\) is negative,\nand two positive numbers can't add to something negative. So both must be **negative**.\n\n- **A** is the opposite conclusion.\n- **C** would make the product negative, contradicting \\(mn > 0\\).\n- **D** (\\(m = -n\\)) would give \\(m + n = 0\\), not less than \\(0\\).\n- **E** would make \\(mn = 0\\), not positive.\n\nThe answer is **B**.", + "hints": [ + "A positive product means the two numbers share a sign.", + "Can two positive numbers sum to a negative? Use that to pick between the two same-sign cases." + ] + }, + { + "id": "quant-number-properties-odds-evens-signs-0019", + "section": "quant", + "topic": "number-properties", + "subtopic": "odds-evens-signs", + "type": "problem-solving", + "difficulty": "medium", + "tags": [ + "parity", + "consecutive-integers", + "must-be-true" + ], + "choices": { + "A": "\\(k^2 + 1\\)", + "B": "\\(k^2 + k\\)", + "C": "\\(2k + 1\\)", + "D": "\\(k^2 + k + 1\\)", + "E": "\\(3k\\)" + }, + "answer": "B", + "author": "openmat", + "reviewers": [], + "status": "in-review", + "prompt": "If \\(k\\) is a positive integer, which of the following is always even?", + "explanation": "Test each choice against both parities of \\(k\\), or use parity rules directly.\n\n- **A. \\(k^2 + 1\\):** if \\(k\\) is even, \\(k^2\\) is even and \\(k^2 + 1\\) is **odd**. Not always even.\n- **B. \\(k^2 + k = k(k+1)\\):** this is the product of two **consecutive** integers, one of which is\n always even. So \\(k(k+1)\\) is **always even**. ✓\n- **C. \\(2k + 1\\):** \\(2k\\) is even, so \\(2k + 1\\) is always **odd**.\n- **D. \\(k^2 + k + 1 = k(k+1) + 1\\):** an even number plus \\(1\\) is always **odd**.\n- **E. \\(3k\\):** has the parity of \\(k\\); if \\(k\\) is odd (e.g. \\(k = 1\\)), \\(3k = 3\\) is **odd**.\n\nOnly **B** is guaranteed even.", + "hints": [ + "Try \\(k = 1\\) and \\(k = 2\\) in each choice and keep only the ones that stay even.", + "Factor \\(k^2 + k\\) — what do you know about the product of two consecutive integers?" + ] + }, + { + "id": "quant-number-properties-odds-evens-signs-0020", + "section": "quant", + "topic": "number-properties", + "subtopic": "odds-evens-signs", + "type": "problem-solving", + "difficulty": "hard", + "tags": [ + "signs", + "product", + "cannot-be-true" + ], + "choices": { + "A": "All three integers are negative.", + "B": "Exactly one of the integers is negative.", + "C": "Exactly two of the integers are negative.", + "D": "One of the integers equals \\(-1\\).", + "E": "The sum of the three integers is positive." + }, + "answer": "C", + "author": "openmat", + "reviewers": [], + "status": "in-review", + "prompt": "The product of three integers is negative. Which of the following CANNOT be true?", + "explanation": "A product is negative only when it is nonzero and the **count of negative factors is odd** — so with\nthree integers, either **one** or **all three** are negative. (None can be zero, or the product would\nbe \\(0\\).)\n\n- **A. All three negative:** three negatives is an odd count → negative product. **Possible.**\n- **B. Exactly one negative:** one negative, two positive → negative product. **Possible.**\n- **C. Exactly two negative:** two negatives is an *even* count, so those two multiply to a positive,\n and the third (nonzero) factor keeps it positive. The product would be **positive**, never negative.\n This **cannot** be true. ✓\n- **D. One equals \\(-1\\):** e.g. \\(-1, 2, 3\\) gives \\(-6\\). **Possible.**\n- **E. Sum is positive:** e.g. \\(-1, 5, 5\\) gives product \\(-25\\) with sum \\(9 > 0\\). **Possible.**\n\nThe answer is **C**.", + "hints": [ + "A negative product needs an *odd* number of negative factors.", + "Multiply out \"exactly two negatives\" with a small example — what sign do you always get?" + ] + }, + { + "id": "quant-number-properties-odds-evens-signs-0021", + "section": "quant", + "topic": "number-properties", + "subtopic": "odds-evens-signs", + "type": "problem-solving", + "difficulty": "medium", + "tags": [ + "parity", + "coefficients", + "must-be-true" + ], + "choices": { + "A": "\\(pq\\)", + "B": "\\(p + q\\)", + "C": "\\(2p + q\\)", + "D": "\\(p + 1\\)", + "E": "\\(pq + 1\\)" + }, + "answer": "B", + "author": "openmat", + "reviewers": [], + "status": "in-review", + "prompt": "If \\(p\\) and \\(q\\) are integers and \\(3p + 5q\\) is even, which of the following must be even?", + "explanation": "An odd coefficient doesn't change parity: \\(3p\\) has the parity of \\(p\\), and \\(5q\\) has the parity of\n\\(q\\). So \\(3p + 5q\\) has the parity of \\(p + q\\). For that to be even, \\(p\\) and \\(q\\) must have the\n**same parity** (both odd or both even).\n\n- **A. \\(pq\\):** both even → even, but both odd → odd. Not guaranteed.\n- **B. \\(p + q\\):** same parity means their sum is **even** every time. This must be even. ✓\n- **C. \\(2p + q\\):** \\(2p\\) is even, so this has the parity of \\(q\\), which could be odd or even.\n- **D. \\(p + 1\\):** flips the parity of \\(p\\); could be either.\n- **E. \\(pq + 1\\):** since \\(pq\\) isn't fixed (see A), neither is \\(pq + 1\\).\n\nThe answer is **B**.", + "hints": [ + "Drop the coefficients: \\(3p + 5q\\) has the same parity as \\(p + q\\).", + "If \\(p + q\\) is even, what does that tell you about whether \\(p\\) and \\(q\\) match in parity?" + ] + }, { "id": "quant-statistics-descriptive-0009", "section": "quant", @@ -984,6 +1134,23 @@ "status": "in-review", "body": "## Overview\n\nPercents and ratios are the most frequently tested arithmetic ideas on GMAT Focus Quant, and\nthey hide inside word problems everywhere. Master three moves: converting between forms, handling\npercent *change*, and scaling ratios.\n\n## Core concepts\n\n**Percent as a factor.** A percent is just a number over 100. The fastest way to apply a percent\nchange is to turn it into a **multiplier**:\n\n- Increase by \\(r\\%\\): multiply by \\(1 + \\tfrac{r}{100}\\). (+25% → \\(\\times 1.25\\))\n- Decrease by \\(r\\%\\): multiply by \\(1 - \\tfrac{r}{100}\\). (−20% → \\(\\times 0.80\\))\n\n**Successive changes multiply.** Two changes in a row are multiplied, never added:\n\n\\[(+25\\%)\\text{ then }(-20\\%): \\quad 1.25 \\times 0.80 = 1.00 \\quad (\\text{no net change})\\]\n\n**Percent change formula.**\n\n\\[\\text{percent change} = \\frac{\\text{new} - \\text{old}}{\\text{old}} \\times 100\\%\\]\n\nAlways divide by the **original** value, not the new one.\n\n**Ratios scale together.** A ratio \\(a : b\\) means the quantities are \\(ak\\) and \\(bk\\) for some\nmultiplier \\(k\\). If boys : girls \\(= 3 : 5\\) and there are 24 boys, then \\(k = 8\\), so there are\n\\(5 \\times 8 = 40\\) girls.\n\n## Worked examples\n\n**Percent change.** A stock rises from $80 to $100. Percent increase \\(= \\tfrac{100 - 80}{80} = \\tfrac{20}{80} = 25\\%\\).\nNote it later falls from $100 back to $80: that's \\(\\tfrac{-20}{100} = -20\\%\\) — a *smaller* percent, because the base is now larger.\n\n**Ratio scaling.** A recipe uses flour : sugar \\(= 7 : 2\\). To use 21 cups of flour, \\(k = 3\\), so\nyou need \\(2 \\times 3 = 6\\) cups of sugar.\n\n## Common traps\n\n- **Adding successive percents.** +25% then −20% is *not* +5%; it's \\(1.25 \\times 0.80 = 1.00\\).\n- **Wrong base.** Percent change always divides by the original amount. A rise then an equal-percent fall does not return to the start.\n- **Ratio ≠ actual count.** \\(3 : 5\\) does not mean 3 and 5 — it means \\(3k\\) and \\(5k\\). Find \\(k\\) first.\n\n## Key takeaways\n\n- Convert percent changes to multipliers and multiply them for successive changes.\n- Percent change = (new − old) / old.\n- A ratio \\(a : b\\) represents \\(ak\\) and \\(bk\\); solve for the multiplier \\(k\\), then scale." }, + { + "id": "quant-number-properties-odds-evens-signs", + "section": "quant", + "topic": "number-properties", + "subtopic": "odds-evens-signs", + "title": "Odds, Evens & Signs", + "tags": [ + "parity", + "odd-even", + "signs", + "number-properties" + ], + "author": "openmat", + "reviewers": [], + "status": "in-review", + "body": "## Overview\n\nMany GMAT Focus Quant questions never ask you to compute a value — they ask what **must** be\ntrue about whether a number is odd or even, or positive or negative. These **parity** and **sign**\nquestions look abstract, but they collapse to a handful of fixed rules. Once you know the rules you\ncan answer in seconds, and when you don't, you can always fall back on **plugging in small numbers**.\n\n## Core concepts\n\n**Parity of sums and differences.** Addition and subtraction have the *same* parity behavior:\n\n\\[\\text{even} \\pm \\text{even} = \\text{even}, \\quad \\text{odd} \\pm \\text{odd} = \\text{even}, \\quad \\text{even} \\pm \\text{odd} = \\text{odd}\\]\n\nThe takeaway: \\(a + b\\) and \\(a - b\\) always have the **same** parity. A sum (or difference) is odd\n**only** when exactly one of the two numbers is odd.\n\n**Parity of products.** A product is even if **any** factor is even; it is odd **only** when **every**\nfactor is odd:\n\n\\[\\text{even} \\times \\text{anything} = \\text{even}, \\qquad \\text{odd} \\times \\text{odd} = \\text{odd}\\]\n\n**Two consecutive integers.** \\(k\\) and \\(k+1\\) are always one odd and one even, so their product\n\\(k(k+1)\\) is **always even**. This is why expressions like \\(k^2 + k = k(k+1)\\) are guaranteed even.\n\n**Signs of products and quotients.** For multiplication and division, only the *count* of negative\nfactors matters:\n\n- An **even** number of negative factors → **positive** result.\n- An **odd** number of negative factors → **negative** result.\n\nSo a product of three numbers is negative exactly when one or three of them are negative — never\nwhen exactly two are.\n\n**Zero is even, and it has no sign.** \\(0\\) is an even integer. It is neither positive nor negative,\nand any product containing a \\(0\\) is \\(0\\) — a case that quietly breaks \"must be positive/negative\"\nclaims, so watch for whether the problem says the numbers are *nonzero*.\n\n## Worked examples\n\n**Parity of an expression.** Is \\(3p + 5q\\) odd or even? The coefficients don't change parity:\n\\(3p\\) has the parity of \\(p\\), and \\(5q\\) has the parity of \\(q\\). So \\(3p + 5q\\) has the parity of\n\\(p + q\\) — it's even exactly when \\(p\\) and \\(q\\) share the same parity.\n\n**Sign of a quotient.** If \\(a < 0\\), \\(b > 0\\), \\(c < 0\\), then \\(\\dfrac{ab}{c}\\) has two negative\ninputs (\\(a\\) and \\(c\\)) — an even count — so the result is **positive**.\n\n## Common traps\n\n- **Assuming subtraction behaves differently from addition.** \\(x - y\\) has the *same* parity as\n \\(x + y\\). If \\(x + y\\) is odd, so is \\(x - y\\).\n- **Forgetting the odd-count rule for signs.** \"Exactly two negatives\" gives a **positive** product,\n not a negative one.\n- **Ignoring zero.** If the problem doesn't say \"nonzero,\" a variable could be \\(0\\) — which is even\n and sign-less, and often kills a \"must be positive\" answer.\n- **Testing only one case.** When you plug in numbers to eliminate a \"must be true\" choice, try both\n parities and both signs; one clean counterexample is enough to reject a choice.\n\n## Key takeaways\n\n- \\(a + b\\) and \\(a - b\\) always share parity; a sum/difference is odd only when exactly one term is odd.\n- A product is odd only if every factor is odd; a single even factor makes it even.\n- Consecutive integers \\(k(k+1)\\) are always even.\n- Sign of a product/quotient depends only on the parity of the *count* of negatives: even count → positive, odd count → negative.\n- Zero is even and sign-less — check whether variables are allowed to be zero." + }, { "id": "quant-statistics-descriptive", "section": "quant",