Problems on Numbers are among the most scoring and logic-driven topics in arithmetic. But many students find them confusing because numbers behave differently under operations; they combine, expand, cancel, or transform depending on the pattern inside the expression.

If you understand these three ideas:
✔ Structure of algebraic expressions
✔ How numbers expand or factorize
✔ How to recognize identity patterns

Then you can solve ANY number-based question effortlessly.

This Problems on Numbers guide covers all formulas, concepts, shortcuts, cases, factorization identities, FAQs, and exam tips, making it the perfect one-stop resource for competitive exams.

Quick Overview: Problems on Numbers Formulas

Concept / Situation	Considered (Meaning of Terms)	Used (Purpose)	Formula (With Meaning of Symbols Inside Row)
Difference of Squares	a, b = any two numbers	To simplify product of binomials	(a + b)(a − b) = a² − b²
Square of Sum	a, b = two numbers/variables	To expand expressions	(a + b)² = a² + b² + 2ab
Square of Difference	a, b = two numbers	To expand or simplify	(a − b)² = a² + b² − 2ab
Square of Three Numbers	a, b, c = three variables	To expand expressions with 3 terms	(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
Sum of Cubes	a, b = two numbers	To factorize cube sum	a³ + b³ = (a + b)(a² − ab + b²)
Difference of Cubes	a, b = two numbers	To factorize cube difference	a³ − b³ = (a − b)(a² + ab + b²)
General Cube Identity	a, b, c = three numbers	To factorize 3-term cube expressions	a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)
Special Case Identity	a, b, c = numbers whose sum is zero	To simplify cube expressions quickly	If a + b + c = 0 → a³ + b³ + c³ = 3abc

Formulas for Problems on Numbers

Below is the full formula explanation section, written in the same style as your Train Passing Object formulas.

1. Difference of Squares Formula (Fast Simplification Pattern)

When multiplying expressions like (a + b)(a − b), the middle terms cancel out automatically.
This formula is the shortcut to simplify such products quickly.

Formula: Difference of Squares

(a+b)(a−b)=a2−b2

Where:

• a = first number
• b = second number

Why this formula works

When expanded:

• (a × a) = a²
• (a × −b) = −ab
• (b × a) = +ab
• (b × −b) = −b²

The middle terms (−ab and +ab) cancel out.

So only:

a2−b2

remains.

Common mistakes students make

• Expanding manually and making sign errors
• Forgetting that middle terms always cancel
• Using this formula even when expressions are NOT in (a + b)(a − b) format

Key Tip:

Whenever two brackets have the same terms but opposite signs, apply this identity instantly.

Example

Simplify:

101 × 99

= (100 + 1)(100 − 1)
= 100² − 1²
= 10000 − 1
= 9999

2. Square of Sum Formula (Three-Term Expansion)

Used when expressions appear as (a + b)².
This identity helps expand without long multiplication.

Formula

(a+b)2 = a2+b2+2ab

Why this formula works

Binomial multiplication gives:

• a²
• ab
• ab
• b²

Combine ab + ab → 2ab

Common mistakes

• Missing the 2ab term
• Incorrectly squaring ab
• Changing sign mistakes

Example

(40 + 3)²
= 40² + 3² + 2(40 × 3)
= 1600 + 9 + 240
= 1849

3. Square of Difference Formula (Negative Sign Variation)

Formula

(a−b)2=a2+b2−2ab

Why this formula works

Middle terms become:

• (−ab) + (−ab) = −2ab

Common errors

• Writing +2ab instead of −2ab
• Forgetting to square both terms

Example

(15 − 2)²
= 15² + 2² − 2(15 × 2)
= 225 + 4 − 60
= 169

4. Square of Three Numbers (Three-Term Expansion)

Formula

(a+b+c)2 = a2+b2+c2+2(ab+bc+ca)

Why this formula works

When expanded, the expression has:

• Three squares
• Three pairwise products

Common mistakes

• Missing any pair product
• Incorrectly combining terms

Example

(1 + 2 + 3)²
= (1² + 2² + 3²) + 2(2 + 6 + 3)
= (1 + 4 + 9) + 2(11)
= 14 + 22
= 36

5. Sum of Cubes Formula (Important Factorization Identity)

Formula

a3+b3 = (a+b)(a2−ab+b2)

Why this formula works

The binomial (a + b) pairs with a specific trinomial which removes mixed ab terms during expansion.

Common mistakes

• Using the wrong trinomial sign
• Forgetting that cube expressions factor into two brackets

Example

Find factors of:

8 + 27
= 2³ + 3³
= (2 + 3)(2² − 2×3 + 3²)
= 5(4 − 6 + 9)
= 5 × 7
= 35

6. Difference of Cubes Formula

Formula

a3−b3 = (a−b)(a2+ab+b2)

Why this formula works

The trinomial changes sign to maintain equality after expansion.

Common mistakes

• Writing minus inside trinomial
• Wrong pairing of signs

Example

125 − 64
= 5³ − 4³
= (5 − 4)(25 + 20 + 16)
= 1 × 61
= 61

7. General Cube Identity (Three-Variable Powerful Identity)

Formula

a3+b3+c3−3abc = (a+b+c)(a2+b2+c2−ab−bc−ca)

Why this formula works

It handles complex three-number cube expressions by splitting them into a binomial and a trinomial.

Common mistakes

• Mixing signs
• Forgetting the −3abc term

Example (Wider Calculation)

Let a = 1, b = 2, c = 3
Compute LHS:
1³ + 2³ + 3³ − 3(1)(2)(3)
= 1 + 8 + 27 − 18
= 18

Now RHS:
(1 + 2 + 3)(1² + 2² + 3² − 1·2 − 2·3 − 3·1)
= 6(1 + 4 + 9 − 2 − 6 − 3)
= 6(3)
= 18

Both match.

8. Special Case Identity 

Condition

a+b+c = 0

Then

a3+b3+c3 = 3abc

Why this works

The entire first bracket (a + b + c) becomes zero in the general cube identity.

Example

Let a = 2, b = −1, c = −1
a + b + c = 0
So
a³ + b³ + c³ = 3abc
= 3(2)(−1)(−1)
= 6

Smart Tips and Practical Tricks for Solving Problems on Numbers

Mastering Problems on Numbers becomes simple when you understand how algebraic expressions behave during expansion and factorization. Most students make mistakes not because formulas are difficult, but because they apply them without identifying the correct pattern. This section breaks down the most important concepts into clear, actionable tips so you can solve questions faster and more accurately.

1. Identify the Pattern Before Using Any Formula

Most number identities follow recognizable patterns. If you detect the form square, cube, sum, or difference, solution becomes automatic.

Common patterns include:

• (a + b)² → Square of sum
• (a − b)² → Square of difference
• a³ ± b³ → Sum/Difference of cubes
• (a + b + c)² → Square of three numbers
• a³ + b³ + c³ − 3abc → General cube identity

Recognizing structure eliminates unnecessary steps and prevents guesswork.

2. Avoid Expanding Expressions Manually

Manually multiplying terms leads to mistakes and takes longer. Identities exist to simplify:

• (a + b)(a − b) = a² − b²
• (a + b)² = a² + b² + 2ab

Using identities reduces multi-step multiplication to one clean formula, improving speed and accuracy.

3. Check for the Special Shortcut: a + b + c = 0

This is one of the most powerful tricks in number system problems.

If

a+b+c = 0

Then

a3+b3+c3 = 3abc

This instantly simplifies large cube expressions and saves significant time in exams.

4. Convert Complicated Numbers Into Identity Form

Rewrite numbers to match patterns:

• 51 = 50 + 1 → (a + b)² pattern
• 99 = 100 − 1 → (a − b)² pattern
• 999 = 1000 − 1 → a³ − b³ pattern

This approach reduces heavy calculations into quick identity-based shortcuts.

5. Analyze the Structure Before Deciding the Identity

For example:

• If you see three variables together → try (a + b + c)² or cube identity
• If you see symmetry → square identity likely applies
• If cubes appear → factor using a³ ± b³ formula

Correctly selecting the identity is more important than memorizing formulas.

6. Double-Check the Signs (+ / −)

One of the most common mistakes:

Students mix up
✔ (a + b)² with (a − b)²
✔ a³ + b³ with a³ − b³

Your identity changes entirely based on the sign.

Always re-check:
✔ Is the middle sign plus or minus?
✔ Are the cubes added or subtracted?
✔ Is the final expression factorizable?

7. Practice Typical Competitive Exam Patterns

Most exam questions follow repetition-based patterns:

• Expand using (a + b)²
• Factorize using a³ − b³
• Use special case a + b + c = 0
• Apply (a + b)(a − b) to simplify

Practicing these patterns increases recognition speed, leading to faster solving during exams.

FAQs About Problems on Numbers

Q1. Why are algebraic identities important in number problems?

They simplify big expressions into smaller ones, helping you avoid long multiplications and reduce calculation mistakes.

Q2. When should I use (a + b)(a − b) = a² − b²?

Whenever two expressions differ only by sign, such as 101×99 → (100+1)(100−1).

Q3. Why are square formulas used so often in number questions?

Many expressions appear in squared form. Using identities expands them instantly without manual calculation.

Q4. How do I know whether to use sum or difference of cubes?

Check the connecting sign:

• “+” → use a³ + b³
• “−” → use a³ − b³

Q5. Why is the identity (a + b + c)² useful?

It simplifies expressions containing three terms, which frequently appear in reasoning and aptitude exams.

Q6. When can I apply the shortcut a³ + b³ + c³ = 3abc?

Only when a + b + c = 0. This condition is essential and must always be checked first.

Q7. Why do students often get identity-based questions wrong?

They fail to recognize the correct pattern and expand terms manually, which leads to errors in signs and multiplication.

Q8. How does recognizing patterns help in number problems?

Patterns indicate the identity to apply. Once the pattern is spotted, the question solves itself in 1–2 steps.

Q9. Why are cube identities used in higher-level exams?

Cube identities handle complex expressions quickly and eliminate large multiplications that would otherwise take long time.

Q10. What is the fastest way to master number identities?

Focus on pattern recognition, use identities instead of manual expansion, and practice frequently asked forms repeatedly.