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Class 10 Mathematics: Real Numbers, Pair of Linear Equations, and AP Activities

Practical solved activities covering prime factorisation, HCF, LCM, and irrational number proofs. Includes graphical solutions and consistency conditions for pairs of linear equations in two variables. Concludes with an introduction to arithmetic progressions using toothpick patterns.

Maths
English
· 8 topics
Checked against the video on 18 Oct 2026

Made by AI from the YouTube video 10 class ಗಣಿತ ಚಟುವಟಿಕೆ ಮತ್ತು ಅದರ ಉತ್ತರಗಳೊಂದಿಗೆ (opens YouTube in a new tab) by ವಿದ್ಯಾಲಯ, then checked against it. All credit for the teaching goes to the creator. Tap a topic’s time to confirm a fact in the video.

Class 10 Mathematics: Real Numbers, Pair of Linear Equations, and AP Activities

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Practical solved activities covering prime factorisation, HCF, LCM, and irrational number proofs. Includes graphical solutions and consistency conditions for pairs of linear equations in two variables. Concludes with an introduction to arithmetic progressions using toothpick patterns.

Factors and Classification of Numbers

Natural numbers from 1 to 30 are tabulated with their total number of factors.
Numbers having more than two factors are classified as composite numbers (such as 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30).
Must knowPrime numbers have exactly two factors: 1 and the number itself.
1 has only one factor and is neither prime nor composite.

Number	Factors	Total Factors
1	1	1
2	1, 2	2
4	1, 2, 4	3
6	1, 2, 3, 6	4
12	1, 2, 3, 4, 6, 12	6
24	1, 2, 3, 4, 6, 8, 12, 24	8

Prime Factorisation and Factor Trees

Any composite number can be uniquely expressed as a product of prime factors irrespective of the order of factors.
The factor tree of 120 breaks down into 2 × 60, then 2 × 30, then 2 × 15, and finally 3 × 5, giving 120 = 2³ × 3 × 5.
Must knowThe prime factorisation of 1250 is 2 × 5⁴ (2 × 5 × 5 × 5 × 5).
The prime factorisation of 3136 is 2⁶ × 7².
The prime factorisation of 1484 is 2² × 7 × 53.

Finding HCF by Listing Factors and Prime Factorisation

The factors of 42 are 1, 2, 3, 6, 7, 14, 21, 42 and of 48 are 1, 2, 3, 6, 8, 12, 16, 24, 48.
Common factors of 42 and 48 are 1, 2, 3, 6, making their highest common factor (HCF) 6.
In prime factorisation, 42 = 2 × 3 × 7 and 48 = 2⁴ × 3; taking common prime factors gives HCF = 2 × 3 = 6.
Must knowFor 72 (2³ × 3²) and 56 (2³ × 7), the common factors are 1, 2, 4, 8, yielding an HCF of 8 (2³).

Finding LCM and Relationship between LCM and HCF

Multiples of 6 and 8 are listed to determine their lowest common multiple (LCM), which is 24.
Prime factorisation method gives LCM of 6 and 8 as 2³ × 3 = 24.
For 12 and 15, prime factorisation gives 12 = 2² × 3 and 15 = 3 × 5; LCM is 60 and HCF is 3.
Must knowFor any two positive integers A and B, Product of the numbers = HCF(A, B) × LCM(A, B).
For A = 98 with HCF = 14 and LCM = 588, the second number B is calculated as (14 × 588) / 98 = 84.

Proof of Irrationality

To prove √2 is irrational, assume on the contrary that √2 is rational, expressed as p/q where p and q are co-prime integers and q ≠ 0.
Squaring both sides gives 2q² = p², which implies 2 divides p², and therefore 2 divides p.
Substituting p = 2m gives 2q² = 4m², or q² = 2m², showing that 2 divides q², and therefore 2 divides q.
Must knowThis proves 2 is a common factor of both p and q, contradicting that p and q are co-prime; hence, √2 is irrational.
For √3 - 5 = p/q, rearranging gives √3 = (p + 5q) / q; since (p + 5q)/q is rational, √3 must be rational, which is a contradiction, proving √3 - 5 is irrational.

Graphical Method for Linear Equations

For equations x + y = 5 and x + 2y = 7, coordinates are plotted on a Cartesian plane with scale 1 cm = 1 unit.
Must knowThe two lines intersect at a single point (3, 2), which represents the unique solution to the system.
For equations x + y = 4 and 2x + 2y = 8, the plotted lines coincide completely, yielding infinitely many solutions (dependent and consistent).
For equations x + y = 6 and x + y = 10, the plotted lines are parallel and never intersect, resulting in no solution (inconsistent).

Ratio Comparison for Linear Equations

The nature of solutions and graphical representation can be determined by comparing coefficients of a1*x + b1*y + c1 = 0 and a2*x + b2*y + c2 = 0.
Must knowIf a1/a2 ≠ b1/b2, the lines are intersecting lines and have exactly one unique solution (consistent).
Must knowIf a1/a2 = b1/b2 ≠ c1/c2, the lines are parallel lines and have no solution (inconsistent).
If a1/a2 = b1/b2 = c1/c2, the lines are coincident lines and have infinitely many solutions (dependent/consistent).

Ratio Condition	Nature of Lines	Algebraic Interpretation
a1/a2 ≠ b1/b2	Intersecting lines	Exactly one unique solution
a1/a2 = b1/b2 ≠ c1/c2	Parallel lines	No solution
a1/a2 = b1/b2 = c1/c2	Coincident lines	Infinitely many solutions

Introduction to Arithmetic Progression (AP)

Triangular shapes formed using sticks show an increasing pattern: figure (i) has 3 sticks, figure (ii) has 5 sticks, figure (iii) has 7 sticks.
The sequence formed by the stick counts is 3, 5, 7, 9, 11, 13, 15.
Must knowEach successive term is obtained by adding a fixed number (2) to the preceding term.

Key terms

Prime Number: A natural number greater than 1 that has exactly two factors, 1 and itself.
Composite Number: A natural number that has more than two factors.
HCF: Highest Common Factor; the greatest number that divides two or more given numbers.
LCM: Lowest Common Multiple; the smallest positive integer divisible by two or more given numbers.
Co-prime Numbers: Numbers that have no common factor other than 1.
Consistent System: A pair of linear equations that has at least one solution (either unique or infinitely many).
Inconsistent System: A pair of linear equations that has no solution because lines are parallel.

Formulas

HCF(A, B) × LCM(A, B) = A × B: Product of two numbers equals the product of their HCF and LCM.

Quick revision

The number 1 is neither a prime nor a composite number.
Prime factorisation of 120 is 2³ × 3 × 5.
Prime factorisation of 1250 is 2 × 5⁴.
The HCF of 42 and 48 is 6.
The LCM of 6 and 8 is 24.
If HCF = 14, LCM = 588, and A = 98, the other number B is 84.
√2 and √3 - 5 are proved to be irrational by contradiction.
If a1/a2 ≠ b1/b2, the pair of linear equations has a unique solution.
If a1/a2 = b1/b2 ≠ c1/c2, the pair of linear equations represents parallel lines with no solution.
If a1/a2 = b1/b2 = c1/c2, the pair of linear equations represents coincident lines with infinite solutions.

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