Number Properties content + multi-part Data Insights support (Two-Part & Table Analysis)

CONTENT_SCHEMA.md

@@ -95,9 +95,37 @@ Put statement (1) and statement (2) in the `## Question` body.
 ### Multi-part answers (two-part, table analysis, etc.)
 
 For types with more than one selectable answer (e.g. `two-part-analysis`, `table-analysis`),
-set `answer` to a compact string describing each part, and lay the options out clearly in the
-body. Example: `answer: "Part1=C; Part2=A"` or `answer: "Row1=Yes; Row2=No; Row3=Yes"`.
-Keep the body unambiguous about what each part refers to.
+add a `parts` map that lists the options for each part, and set `answer` to a compact
+`Key=Value` string (parts separated by `;`). The study site renders one selector per part and
+grades them together — the question is correct only if **every** part matches.
+
+```yaml
+parts:
+  Nuts: "$1|$2|$3|$4|$5|$6"
+  Juice: "$1|$2|$3|$4|$5|$6"
+answer: "Nuts=$2; Juice=$3"
+```
+
+```yaml
+parts:
+  Claim1: "Yes|No"
+  Claim2: "Yes|No"
+  Claim3: "Yes|No"
+answer: "Claim1=No; Claim2=Yes; Claim3=Yes"
+```
+
+Rules for `parts`:
+
+- **Part keys** (the labels shown above each selector) must be alphanumeric — use `Nuts`,
+  `Claim1`, `Row1`, etc. (letters, digits, `-`, `_`; **no spaces**). Number claims in the body
+  (e.g. "**Claim 1.** …") so the short key stays clear.
+- **Options** for each part are a single string of choices separated by `|`.
+- **`answer`** matches part keys to the correct option value: `Key=Value; Key2=Value2`. Matching is
+  case-insensitive and trimmed, so the answer values must be spelled the same as the options.
+- Lay the scenario, table, and what each part refers to out clearly in the `## Question` body.
+
+If you omit `parts`, you can still fall back to a plain compact `answer` string (e.g.
+`answer: "Part1=C; Part2=A"`) for a browse-only question, but it will not be gradeable in Practice.
 
 ---
 

content/lessons/data-insights/data-insights-table-analysis-ratios-thresholds.md

@@ -0,0 +1,64 @@
+---
+id: data-insights-table-analysis-ratios-thresholds
+section: data-insights
+topic: table-analysis
+subtopic: ratios-thresholds
+title: "Table Analysis: Ratios & Thresholds"
+tags: [table-analysis, percent-change, ratios, thresholds]
+author: openmat
+reviewers: []
+status: in-review
+original: true
+license: CC-BY-SA-4.0
+---
+
+## Overview
+
+Table Analysis presents a sortable table and a set of **Yes/No** (or True/False) statements you
+must judge one at a time. Each row of statements is graded independently, so a single question is
+really several mini-questions. The skill is reading the table precisely and testing each claim with
+the least arithmetic possible.
+
+## Core concepts
+
+**Read the claim's quantifier first.** "Every," "at least one," "the greatest," "more than" — the
+quantifier tells you what evidence settles the claim. An "every"/"all" claim is disproved by **one**
+counterexample; a "there exists" claim is proved by one supporting case. Find that one case fast.
+
+**Percent change is measured against the original.** For a jump from an old value to a new value,
+
+\[\text{percent change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%.\]
+
+Always divide by the **old** value, not the new one — the most common table-analysis error.
+
+**Turn a threshold into a target number.** Instead of computing an exact percent, convert "more than
+40% above 90" into a concrete bar: 40% of 90 is 36, so the target is \(90 + 36 = 126\). Now you just
+compare: is the new value above 126? This is faster and less error-prone than dividing.
+
+**Only compute what the claim needs.** You rarely need every cell. For "greatest two-quarter total,"
+sum each store's two columns and compare — ignore everything else.
+
+## Worked examples
+
+Using a table with Q1/Q2 units — Alpha (120, 150), Beta (200, 180), Gamma (90, 140):
+
+**"Every store grew in Q2." → No.** Beta fell from 200 to 180. One counterexample settles it; you
+needn't check the others once you find it.
+
+**"Beta had the greatest two-quarter total." → Yes.** Totals: 270, 380, 230. Beta's 380 wins.
+
+**"Gamma rose more than 40% from Q1 to Q2." → Yes.** Threshold: 40% of 90 = 36, target 126. Since
+\(140 > 126\), the increase clears 40%. (Exact: \(50/90 \approx 55.6\%\).)
+
+## Common traps
+
+- **Dividing by the new value** when finding percent change — use the original as the base.
+- **Over-computing.** Don't calculate every percent; convert thresholds to target numbers and compare.
+- **Treating the statements as linked.** Each Yes/No row stands alone — judge it on its own evidence.
+
+## Key takeaways
+
+- Let the quantifier ("every," "at least one," "greatest") tell you what evidence to look for.
+- Percent change divides by the **original** value.
+- Convert a percent threshold into a concrete target number, then just compare.
+- Evaluate each statement independently and compute only what that statement requires.

content/lessons/data-insights/data-insights-two-part-analysis-simultaneous-conditions.md

@@ -0,0 +1,60 @@
+---
+id: data-insights-two-part-analysis-simultaneous-conditions
+section: data-insights
+topic: two-part-analysis
+subtopic: simultaneous-conditions
+title: "Two-Part Analysis: Simultaneous Conditions"
+tags: [two-part-analysis, system-of-equations, simultaneous-conditions]
+author: openmat
+reviewers: []
+status: in-review
+original: true
+license: CC-BY-SA-4.0
+---
+
+## Overview
+
+Two-Part Analysis questions give you a single scenario and ask for **two** answers — one per
+column — chosen from a shared list of options. The two answers are usually linked, so the trap is
+picking a value that satisfies one condition while quietly breaking the other. Treat the two
+columns as **one system** to solve, not two independent questions.
+
+## Core concepts
+
+**Translate each condition into an equation or inequality.** Most two-part prompts hand you two
+constraints. Name a variable for each column and write one relation per constraint. Two linked
+constraints in two unknowns almost always pin down a unique pair.
+
+**Solve the system, don't guess-and-check one column.** With two linear equations, use elimination
+(add or subtract to cancel a variable) or substitution. A pair that fits only the first equation is
+exactly the distractor the question is built around.
+
+**Watch what each column actually asks for.** The columns can request different things — a value
+and a rate, a "before" and an "after," a minimum and a maximum. Read the column headers carefully;
+the two answers are not always the same *kind* of quantity.
+
+**Verify against every condition.** Before committing, plug your two answers back into **all** the
+constraints, including the one you didn't use to solve. This single habit catches most two-part
+mistakes.
+
+## Worked examples
+
+**A linked system.** Two receipts: \(2n + j = 7\) and \(n + 2j = 8\). Adding gives
+\(3n + 3j = 15\), so \(n + j = 5\); subtracting that from the first gives \(n = 2\), hence \(j = 3\).
+Check the second equation: \(2 + 2(3) = 8\). ✓ Column 1 = 2, Column 2 = 3.
+
+**Guarding against a partial fit.** In the example above, \((n, j) = (3, 1)\) satisfies the first
+receipt (\(2(3) + 1 = 7\)) but not the second (\(3 + 2 = 5 \ne 8\)). Only a pair that clears **both**
+conditions is correct.
+
+## Common traps
+
+- **Solving one column in isolation.** The columns are coupled — find the pair, not two separate answers.
+- **Grabbing the first pair that fits one equation.** Always test against the other condition too.
+- **Mixing up the columns.** Confirm which quantity each column wants before you select.
+
+## Key takeaways
+
+- Assign one variable per column and turn each condition into an equation or inequality.
+- Solve the constraints as a single system (elimination/substitution), then verify against all of them.
+- The classic distractor satisfies one condition but not the other — a full check kills it.

content/lessons/quant/quant-number-properties-odds-evens-signs.md

@@ -0,0 +1,76 @@
+---
+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, positive-negative]
+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 an expression is odd or even, or positive or negative. These are **parity** and
+**sign** questions. You almost never need the actual numbers; you only need to track two-state
+information (odd/even, +/−) through the arithmetic. Learn the rules and you can answer in seconds.
+
+## Core concepts
+
+**Parity rules (odd/even).** An even number is a multiple of 2; an odd number is not. The only
+rules you need:
+
+- **Addition/subtraction:** the result is odd exactly when you combine one odd and one even.
+  even ± even = even, odd ± odd = even, even ± odd = **odd**.
+- **Multiplication:** a product is even if *any* factor is even. It is odd only when *every*
+  factor is odd. So \(\text{even} \times \text{anything} = \text{even}\).
+
+A useful consequence: \(n(n+1)\), the product of two **consecutive** integers, is always even —
+one of any two consecutive integers is even. Likewise \(n^2\) has the same parity as \(n\).
+
+**Sign rules (positive/negative).** For a product or quotient, only the *count of negative
+factors* matters:
+
+- An **even** number of negative factors → the result is **positive**.
+- An **odd** number of negative factors → the result is **negative**.
+
+Zero is neither positive nor negative, and any product containing a zero factor is zero. Watch
+for it — "\(xy > 0\)" quietly tells you neither \(x\) nor \(y\) is 0, while "\(xy \ge 0\)" does not.
+
+**Recovering one sign from a product.** If you know the sign of a whole product and the signs of
+all but one factor, you can solve for the last one. If \(abc > 0\) and \(bc < 0\), then
+\(a = \dfrac{abc}{bc} = \dfrac{(+)}{(-)} < 0\).
+
+## Worked examples
+
+**Parity of an expression.** Is \(n^2 + n + 1\) odd or even for every integer \(n\)?
+\(n^2 + n = n(n+1)\) is a product of consecutive integers, so it is always even. Adding 1 makes
+the whole expression **always odd** — no need to test cases.
+
+**Signs from constraints.** If \(x < 0\) and \(xy^2 z > 0\), what is the sign of \(z\)?
+Since \(y^2 \ge 0\) and the product is nonzero, \(y^2 > 0\) (positive). The product's sign is
+\((\text{sign of } x)(\text{sign of } y^2)(\text{sign of } z) = (-)(+)(\text{sign of } z) > 0\),
+so \(z\) must be **negative**.
+
+## Common traps
+
+- **Assuming a variable is an integer.** Parity rules apply only to integers. If a problem
+  doesn't say "integer," \(x\) could be \(2.5\) and "odd/even" is meaningless.
+- **Forgetting even × odd = even.** A product is even as soon as *one* factor is even; you do not
+  need both.
+- **Losing zero.** \(x^2 \ge 0\), not \(> 0\), unless you know \(x \ne 0\). A stray zero flips a
+  "must be positive" into "could be zero."
+- **Squares don't erase parity.** \(n^2\) is odd when \(n\) is odd. Squaring changes the sign
+  situation (never negative) but not the parity.
+
+## Key takeaways
+
+- Sum/difference is odd only when mixing one odd and one even; a product is even whenever any
+  factor is even.
+- \(n(n+1)\) is always even; \(n^2\) has the same parity as \(n\).
+- For products/quotients, count the negatives: even count → positive, odd count → negative.
+- Track only the two-state information (odd/even, +/−); you rarely need the actual values.

content/lessons/quant/quant-number-properties-remainders.md

@@ -0,0 +1,73 @@
+---
+id: quant-number-properties-remainders
+section: quant
+topic: number-properties
+subtopic: remainders
+title: "Remainders"
+tags: [remainders, division, modular-arithmetic, cyclicity]
+author: openmat
+reviewers: []
+status: in-review
+original: true
+license: CC-BY-SA-4.0
+---
+
+## Overview
+
+Remainder questions look intimidating but reduce to one equation and a few habits. The GMAT loves
+them because they reward structure over brute force: you can answer "what is the remainder when
+\(3^{47}\) is divided by 5?" without ever computing \(3^{47}\).
+
+## Core concepts
+
+**The division identity.** When a positive integer \(n\) is divided by a positive integer \(d\),
+there is a unique quotient \(q\) and remainder \(r\) with
+
+\[n = qd + r, \qquad 0 \le r < d.\]
+
+The remainder is always at least 0 and strictly less than the divisor. Whenever a problem says
+"the remainder is \(r\) when divided by \(d\)," immediately write \(n = qd + r\) — that single
+substitution unlocks most questions.
+
+**Remainders add and multiply.** You can do arithmetic on remainders directly (this is modular
+arithmetic). If \(n\) leaves remainder \(r\) on division by \(d\), then to find the remainder of
+\(an + b\), just compute \(ar + b\) and reduce mod \(d\). Substituting \(n = qd + r\) shows why:
+the \(qd\) part is divisible by \(d\) and contributes nothing to the remainder.
+
+**Cyclicity of powers.** Remainders of \(a^1, a^2, a^3, \dots\) on division by \(d\) eventually
+repeat in a fixed-length cycle. Find the cycle, then use the exponent's position within it. For
+\(3^k \bmod 5\): the remainders are \(3, 4, 2, 1, 3, 4, 2, 1, \dots\) — a cycle of length 4. To get
+\(3^{47}\), note \(47 = 4(11) + 3\), so \(3^{47}\) sits at position 3 in the cycle → remainder \(2\).
+
+**Combining two conditions.** "Leaves remainder 2 mod 5 and remainder 1 mod 3" defines a single
+family of numbers. List the smaller list and check it against the other condition:
+\(\{2, 7, 12, 17, 22, \dots\}\) against "\(\equiv 1 \bmod 3\)" first hits at \(7\), and then every
+\(15\) (\(= 5 \times 3\)) after that: \(7, 22, 37, \dots\).
+
+## Worked examples
+
+**Linear expression.** When \(n\) is divided by 7 the remainder is 4. What is the remainder when
+\(3n + 5\) is divided by 7? Work with the remainder: \(3(4) + 5 = 17\), and \(17 = 2(7) + 3\), so
+the remainder is **3**. (Check with \(n = 11\): \(3(11)+5 = 38 = 5(7)+3\). ✓)
+
+**Power.** Remainder of \(7^{100}\) mod 4? \(7 \equiv 3 \bmod 4\), and powers of 3 mod 4 cycle
+\(3, 1, 3, 1, \dots\) (length 2). Even exponent → position 2 → remainder **1**.
+
+## Common traps
+
+- **Remainder ≥ divisor.** A remainder must be less than the divisor. If your work gives a
+  remainder of 17 when dividing by 7, reduce it (\(17 \to 3\)).
+- **Multiplying then forgetting to reduce.** \(3 \times 4 + 5 = 17\) is not the answer mod 7 —
+  reduce it to 3.
+- **Assuming a unique number.** "Remainder 3 when divided by 5" describes infinitely many numbers
+  (\(3, 8, 13, \dots\)); pick the smallest that fits, or keep it as \(5q + 3\).
+- **Miscounting the cycle position.** If the cycle has length 4 and the exponent is a multiple of
+  4, you are at the *end* of the cycle (position 4), not position 0.
+
+## Key takeaways
+
+- Write \(n = qd + r\) with \(0 \le r < d\) the moment a remainder is mentioned.
+- Do arithmetic on the remainders themselves, then reduce mod \(d\).
+- For powers, find the repeating cycle of remainders and use the exponent's position in it.
+- For two simultaneous remainder conditions, list one and test against the other; solutions repeat
+  every \(\text{lcm}\) of the divisors.

content/questions/data-insights/data-insights-table-analysis-ratios-thresholds-0010.md

@@ -0,0 +1,62 @@
+---
+id: data-insights-table-analysis-ratios-thresholds-0010
+section: data-insights
+topic: table-analysis
+subtopic: ratios-thresholds
+type: table-analysis
+difficulty: medium
+tags: [table-analysis, percent-change, thresholds]
+parts:
+  Claim1: "Yes|No"
+  Claim2: "Yes|No"
+  Claim3: "Yes|No"
+answer: "Claim1=No; Claim2=Yes; Claim3=Yes"
+author: openmat
+reviewers: []
+status: in-review
+original: true
+license: CC-BY-SA-4.0
+---
+
+## Question
+
+The table shows units sold by three stores in the first two quarters of a year.
+
+| Store | Q1 units | Q2 units |
+|-------|---------:|---------:|
+| Alpha |      120 |      150 |
+| Beta  |      200 |      180 |
+| Gamma |       90 |      140 |
+
+For each claim, select **Yes** if it is supported by the table and **No** if it is not.
+
+- **Claim 1.** Every store sold more units in Q2 than in Q1.
+- **Claim 2.** Beta had the greatest two-quarter total (Q1 + Q2) of the three stores.
+- **Claim 3.** Gamma's Q2 units were more than 40% above its Q1 units.
+
+## Explanation
+
+Work one claim at a time; each is a quick check, not a calculation marathon.
+
+**Claim 1 — No.** Alpha (120 → 150) and Gamma (90 → 140) rose, but **Beta fell** from 200 to 180.
+A single counterexample makes an "every" statement false.
+
+**Claim 2 — Yes.** Compute the two-quarter totals: Alpha \(= 120 + 150 = 270\), Beta \(= 200 + 180 = 380\),
+Gamma \(= 90 + 140 = 230\). Beta's 380 is the largest.
+
+**Claim 3 — Yes.** Gamma went from 90 to 140, an increase of 50. As a percent of the original,
+
+\[\frac{50}{90} \approx 55.6\% ,\]
+
+which is more than 40%. (Shortcut: 40% of 90 is 36, so 90 + 36 = 126; since 140 > 126, the increase
+clears the 40% threshold.)
+
+So the answers are **No, Yes, Yes**.
+
+The trap on Claim 3 is dividing by the wrong base — percent change is always measured against the
+**original** value (90), not the new one (140).
+
+## Hints
+
+- For "every"-type claims, hunt for one counterexample before checking the rest.
+- Percent increase = (new − old) / **old**. Or compare against a threshold: 40% of 90 is 36.