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Problems on Divisibility - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • 13 Questions around this concept.

Solve by difficulty

QuestionMEDIUM

The number 101^{100}-1 is divisible by 

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For every natural number  n, 3^{2 n+2}-8 n-9  is divisible by:

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(1+x)^n-n x-1  is divisible by (where n \in N ):

 

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The greatest integer, which divides the number  (101^{100}-1) is

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49^n+16 n-1 is divisible by:

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The difference between an integer and its cube is divisible by    

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Concepts Covered - 2

Problems on Divisibility

Given Expression is divisible by an Integer

\\\mathrm{1.\;\;Expression,\;(1+x)^n-1\;is\;divisible\;by\;x\;because}\\\mathrm{\;\;\;\;\;(1+x)^n-1= \;(^n C_0\;+\;^nC_1x\;+\;^nC_2x^2+......+^nC_nx^n)-1}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=x\left [ \;\;^nC_1\;+\;^nC_2x+......+^nC_nx^{n-1} \right ]}\\\mathrm{2.\;\;Expression,\;(1+x)^n-nx-1\;is\;divisible\;by\;x^2\;because}\\\mathrm{\;\;\;\;\;(1+x)^n-nx-1= \;(^n C_0\;+\;^nC_1x\;+\;^nC_2x^2+......+^nC_nx^n)-nx-1}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=x^2\left [ \;\;^nC_2+\;^nC_3x+......+^nC_nx^{n-2} \right ]}


 

For example: 

\\Prove\,\,that\,\,3^{2 n+2}-8 n-9 \text { is divisible by } 8 \text { if } n \in N \\\\ \begin{array}{l}{\begin{aligned} 3^{2 n+2} &-8 n-9=(1+8)^{n+1}-8 n-9 \\ &=\left[1+(n+1) 8+^{n+1} C_{2} 8^{2}+\ldots\right]-8 n-9 \\ &=^{n+1} C_{2} 8^{2}+^{n+1} C_{3} 8^{3}+^{n+1} C_{4} 8^{4}+\ldots \\ &=8\left[^{n+1} C_{2}8+^{n+1} C_{3} 8^2+^{n+1} C_{4} 8^{3}+\ldots .\right] \end{aligned}} \\ {\text { which is clearly divisible by } 8 .}\end{array}

Some standard results of Divisibility Problem

Divisibility: Important Results

  1. The expression \mathrm{a^n-b^n} is divisible by \mathrm{a+b}, if \mathrm{n} is even.

  2. The expression \mathrm{a^n-b^n} is divisible by \mathrm{a-b}, if n is even or odd.

  3. The expression \mathrm{a^n+b^n} is divisible by , \mathrm{a+b} if n is odd.

In all the above cases n is a natural numbers.

For Example

The expression \mathrm{15^4-7^4} is divisible by (15+7)=22 \text { and }(15-7)=8.

Study it with Videos

Problems on Divisibility
Some standard results of Divisibility Problem

Study other Related Concepts

Problems on Divisibility Current Topic

Binomial Theorem - Formula, Expansion, Problems and Applications
Some Standard Expansions
General and Middle Terms
Greatest Term (numerically)
An Important Theorem
Problems on Divisibility
Finding last digits
Important Result (Comparison)
Multinomial Theorem
Series Involving Binomial Coefficients

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