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

Algebraic Word Problems and Number Properties notes for XAT: 28 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Algebraic Word Problems and Number Properties

Chapter Roadmap

Step 1: Number Properties & Algebraic Expressions

Master parity, divisibility, prime constraints, symmetric identities, and the behavior of exponents in integer domains.

Step 2: Algebraic Word Problems & Data Sufficiency

Translate real-world scenarios into equations, optimize variables, and rigorously test if given statements provide a unique solution.

Topic Hero: Number Properties & Algebraic Expressions

The Core Intuition

Algebra in competitive exams is about understanding the behavior of numbers and manipulating expressions to reveal hidden constraints.

The Golden Rule of Domains

Integers? Look for factor pairs and divisibility.

Primes? Look for restricted factorizations.

Positive? Eliminate negative roots and zero.

First-Principles Clarity:
Constraints are not obstacles; they are the primary filters that narrow down infinite real solutions to a single integer answer.

Concept: Exponents and Perfect Powers

The Perfect Power Rule

For an expression composed of prime bases to be a perfect $k$ -th power, the exponent of every prime factor must be a multiple of $k$ .

Example: If

3 3^{A} \cdot 7^{B}

is a positive integer, then:

$A$ must be a multiple of 3.
$B$ must be a multiple of 3.
Both $A \geq 0$ and $B \geq 0$ (otherwise, the result is a fraction).

Key Takeaway:
Perfect integer powers require non-negative exponents that are perfectly divisible by the root index.

Concept: Symmetric Expressions and Reciprocals

The Reciprocal Shortcut

When given the sum of $n$ -th powers and the sum of their reciprocals, you can find the product of the variables directly.

Derivation:
Given:

x^{n} + y^{n} = S

and

x^{- n} + y^{- n} = R

\frac{1}{x ^{n}} + \frac{1}{y ^{n}} = \frac{x ^{n} + y ^{n}}{( x y ) ^{n}} = R

\frac{S}{( x y ) ^{n}} = R ⟹ (x y)^{n} = \frac{S}{R}

Exam Application:
Use this to bypass finding

x

and

y

individually when asked for

(x y)^{k}

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 notes in this unit

chapter

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

Algebraic Word Problems and Number Properties notes for XAT: 28 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questi

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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