Prove that the square of any positive integer is of the form 3m or, 3m + 1 but not of the form 3m + 2.
To Prove: that the square of an positive integer is of the form 3m or 3m + 1 but not of the form 3m + 2.
Proof: Since positive integer n is of the form of 3q, 3q + 1 and 3q + 2
If n = 3q
$\Rightarrow n^{2}=(3 q)^{2}$
$\Rightarrow n^{2}=9 q^{2}$
$\Rightarrow n^{2}=3\left(3 q^{2}\right)$
$\Rightarrow n^{2}=3 m\left(m=3 q^{2}\right)$
If n = 3q + 1
Then, $n^{2}=(3 q+1)^{2}$
$\Rightarrow n^{2}=\left(3 q^{2}\right)+6 q+1$
$\Rightarrow n^{2}=9 q^{2}+6 q+1$
$\Rightarrow n^{2}=3 q(3 q+1)+1$
$\Rightarrow n^{2}=3 m+1($ where $m=(3 q+2))$
If $n=3 q+2$
Then, $n^{2}=(3 q+2)^{2}$
$\Rightarrow n^{2}=\left(3 q^{2}\right)+12 q+4$
$\Rightarrow n^{2}=9 q^{2}+12 q+4$
$\Rightarrow n^{2}=3(3 q+4 q+1)+1$
$\Rightarrow n^{2}=3 m+1$ (where $q=(3 q+4 q+1)$ )
Hence n2 integer is of the form 3m, 3m + 1 but not of the form 3m + 2.
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,