Algebraic Word Problems and Number Properties Short Notes for XAT: Concepts, Formulas, Worked Examples & Practice

Algebraic Word Problems and Number Properties short notes for XAT: 2 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Summary: Quick Recall for Number Properties

Quick Recall

Core Algebraic Identities

Reciprocal Shortcut: $(x y)^{n} = \frac{x ^{n} + y ^{n}}{x ^{- n} + y ^{- n}}$
Difference of Squares: $x^{2} - y^{2} - p (x + y) = (x + y) (x - y - p)$

Number Theory Hooks

Perfect $k$ -th Power: Exponents must be non-negative multiples of $k$ .
Prime Constraints: $A \times B = k \cdot p ⟹$ pairs are ${1, p, k, k p}$ .
Domain Filters: "Positive integers" $⟹$ factors $> 0$ , sums $>$ differences.

Geometry-Algebra Bridge

Area Ratio: Ratio of the squares of their algebraic roots ( $r_{1}^{2} : r_{2}^{2} : r_{3}^{2}$ ).

Final Checklist for Word Problems & DS

Final Checklist

Word Problems

Define Bounds: Integers? Positive? Digits (0-9)?
Translate Carefully: Check order for "less than".
Digit Rules: Use $10 a + b$ . Reversed diff is multiple of 9.
Scoring: Substitute $u = N - c - w$ to reduce variables.

Data Sufficiency

Golden Rule: Yields exactly ONE unique answer?
Strict Order: Test I $\to$ Test II $\to$ Combine.
Beware Trap: Finding a solution $\neq =$ sufficient.
Check Constraints: Apply implicit bounds first.

Algebraic Word Problems and Number Properties: Solved Questions with Step-by-Step Explanations (2 Problems)

Question 1 · Quantitative Aptitude and Data Interpretation (QA & DI) NAT

Let $n$ be a positive integer such that $n^{3} + 2 n^{2} + 3 n + 4$ is a perfect cube. What is the sum of all possible values of $n$ ?

Correct Answer:

Step-by-Step Solution

Key idea: This is a perfect power bounding problem. For large $n$ , the expression lies strictly between consecutive cubes, so only small $n$ need checking. Modular arithmetic eliminates remaining candidates.

Step 1: Compare with nearby cubes. Note $(n + 1)^{3} = n^{3} + 3 n^{2} + 3 n + 1$ .

Our expression: $E (n) = n^{3} + 2 n^{2} + 3 n + 4$ .

Difference: $(n + 1)^{3} - E (n) = n^{2} - 3$ .

For $n \geq 2$ , $n^{2} - 3 > 0 \Rightarrow E (n) < (n + 1)^{3}$ .

Step 2: Compare with $n^{3}$ . Clearly $E (n) > n^{3}$ for $n > 0$ .

So for $n \geq 2$ , $n^{3} < E (n) < (n + 1)^{3}$ . Thus $E (n)$ cannot be a perfect cube for $n \geq 2$ .

Step 3: Check $n = 1$ : $E (1) = 1 + 2 + 3 + 4 = 10$ , not a cube.

Step 4: No positive integer $n$ satisfies the condition. Sum of all possible values is 0.

Step 5: Verify no edge cases missed. For $n = 0$ (not positive), $E (0) = 4$ , not cube. Negative $n$ excluded by problem.

Answer: 0

Question 2 · Quantitative Aptitude and Data Interpretation (QA & DI) MCQ

Let $N$ be the smallest positive integer that has exactly three distinct prime factors and exactly 24 positive divisors. A rectangle has integer side lengths $x$ and $y$ such that its area is numerically equal to $N$ times its semi-perimeter. How many unordered pairs of side lengths ${x, y}$ exist such that $g cd (x - N, y - N) = 1$ ?

A.
2
B.
4
C.
8
D.
12

Correct Answer:

Step-by-Step Solution

Key idea: This is a Simon's Favorite Factoring Trick problem combined with divisor counting and coprimality. The geometric condition translates to a Diophantine equation, which we then factor.

Step 1: Find $N$ . We need the smallest integer with 3 distinct prime factors and 24 divisors.

Let $N = p_{1}^{a} p_{2}^{b} p_{3}^{c}$ . The number of divisors is $(a + 1) (b + 1) (c + 1) = 24$ .

To minimize $N$ , assign the largest exponents to the smallest primes (2, 3, 5).

The optimal factorization of 24 into three integers $> 1$ is $4 \times 3 \times 2$ .

Thus, $a = 3, b = 2, c = 1 ⟹ N = 2^{3} \cdot 3^{2} \cdot 5^{1} = 8 \cdot 9 \cdot 5 = 360$ .

Step 2: Translate the geometric condition. Area $= x y$ , semi-perimeter $= x + y$ .

$x y = N (x + y) ⟹ x y - N x - N y = 0$ .

Add $N^{2}$ to both sides: $x y - N x - N y + N^{2} = N^{2} ⟹ (x - N) (y - N) = N^{2}$ .

Step 3: Apply the coprimality constraint. Let $u = x - N$ and $v = y - N$ . We need $uv = N^{2}$ and $g cd (u, v) = 1$ .

Since $N = 2^{3} \cdot 3^{2} \cdot 5$ , $N^{2} = 2^{6} \cdot 3^{4} \cdot 5^{2}$ .

For $u$ and $v$ to be coprime, they cannot share any prime factors. Thus, each prime power ( $2^{6}, 3^{4}, 5^{2}$ ) must be assigned entirely to either $u$ or $v$ .

Step 4: Count the pairs. There are 3 distinct prime powers, so there are $2^{3} = 8$ ordered pairs $(u, v)$ .

Since $N^{2} > 1$ and $g cd (u, v) = 1$ , $u \neq = v$ . Thus, the number of unordered pairs ${u, v}$ is $8/2 = 4$ .

Each valid ${u, v}$ gives a unique valid ${x, y}$ since $x, y > N$ .

Answer: 4

More short notes in this unit

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Algebraic Word Problems and Number Properties Short Notes for XAT: Concepts, Formulas, Worked Examples & Practice

Algebraic Word Problems and Number Properties short notes for XAT: 2 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice q

A question from this chapter

Question 1

Let n be a positive integer such that n^{3} + 2 n^{2} + 3 n + 4 is a perfect cube. What is the sum of all possible values of n ?

Question 2

Let N be the smallest positive integer that has exactly three distinct prime factors and exactly 24 positive divisors. A rectangle has integer side lengths x and y such that its area is numerically equal to N times its semi-perimeter. How many unordered pairs of side lengths {x, y} exist such that g cd (x - N, y - N) = 1 ?

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