Tag: fundamental theorem of arithmetic

Questions Related to fundamental theorem of arithmetic

Say true or false:
A positive integer is of the form $3q + 1,$ $q$  being a natural number, then you write its square in any form other than  $3m + 1$, i.e.,$ 3m $ or $3m + 2$  for some integer $m$.

maths real number fundemental theorem of arithmetic real numbers on number line fundamental theorem of arithmetic

  1. True

  2. False

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Correct Option: B
Explanation:

Let the positive integer $n$ is of the form $3q, 3q+1,$ and $ 3q+2$
If $n=3q$
Squaring both sides, we get,
    $=>{ n }^{ 2 }=9{ q }^{ 2 }$
    $=>{ n }^{ 2 }=3\left( { 3q }^{ 2 } \right) $
    $=>{ n }^{ 2 }=3m$, where $m=3{ q }^{ 2 }$
Now, if $n=3q+1$
    $=>{ n }^{ 2 }={ \left( 3q+1 \right)  }^{ 2 }$
    $=>{ n }^{ 2 }=9{ q }^{ 2 }+6q+1$
    $=>{ n }^{ 2 }={ 3q\left( 3q+2 \right)  }+1$
    $=>{ n }^{ 2 }=3m+1 ,$ where $  m=q\left( 3q+2 \right) $
Now, if $n=3q+2$
    $=>{ n }^{ 2 }={ \left( 3q+2 \right)  }^{ 2 }$
    $=>{ n }^{ 2 }=9{ q }^{ 2 }+12q+4$
    $=>{ n }^{ 2 }={ 3q\left( 3q+4 \right)  }+4$
    $=>{ n }^{ 2 }={ 3q\left( 3q+4 \right)  }+3+1$
    $=>{ n }^{ 2 }=3m+1$ where $m=\left( 3{ q }^{ 2 }+4q+1 \right) $
Hence, ${ n }^{ 2 }$ integer is of the form $3m$ and $3m+1$ not $3m+2$

A rectangular veranda is of dimension $18$m $72$cm $\times 13$ m $20$ cm. Square tiles of the same dimensions are used to cover it. Find the least number of such tiles.

maths real number real numbers on number line fundamental theorem of arithmetic common factors and hcf

  1. $4290$

  2. $4540$

  3. $4620$

  4. $4230$

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Correct Option: A
Explanation:

The edge of rectangular veranda are $18\ m\ 72\ cm=1872\ cm$ and $13\ m\ 20\ cm=1320\ cm$.


On taking $HCF$ of $1872$ and $1320$, we get

$HCF=24$

Therefore,
No. of tiles required $=$ $\dfrac{Area\ of\ Veranda}{Area\ of\ tiles}$

                                  $=\dfrac{1872\times 1320}{24\times 24}$

                                  $=4290$

Hence, this is the answer.

What is the H.C.F. of two co-prime numbers ?

maths real number real numbers on number line fundamental theorem of arithmetic common factors and hcf

  1. $1$

  2. $0$

  3. $2$

  4. none of these

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Correct Option: A
Explanation:

The two numbers which have only 1 as their common factor are called co-primes.

For example, Factors of $ 5 $  are $ 1, 5 $
Factors of $ 3 $ are $ 1, 3 $

Common factors is $ 1 $.
$ => HCF = 1 $

The HCF of $256,442$ and $940$ is

maths real number real numbers on number line fundamental theorem of arithmetic common factors and hcf

  1. $2$

  2. $14$

  3. $142$

  4. none

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Correct Option: A
Explanation:

Prime factors of numbers are 

$256=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\ 442=2\times 13\times 17\ 940=2\times 2\times 5\times 47\ Hence,\quad HCF=2$
So, correct answer is option A. 

HCF of $x^2 -y^2$ and $x^3-y^3$ is

maths real number real numbers on number line fundamental theorem of arithmetic common factors and hcf

  1. $x-y$

  2. $x^3-y^3$

  3. $(x^2-y^2)$

  4. $(x+y)(x^2+xy+y^2)$

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Correct Option: A
Explanation:

Since, $x^2-y^2=(x+y)(x-y)$
$x^3-y^3=(x-y)(x^2+xy+y^2)$
$H.C.F.=(x-y)$
Option A is correct.

Determine the HCF of $a^2 - 25, a^2 -2a -35$ and $a^2+12a+35$

maths real number real numbers on number line fundamental theorem of arithmetic common factors and hcf

  1. (a-5)(a+7)

  2. (a+5)(a-7)

  3. (a-7)

  4. (a+5)

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Correct Option: D
Explanation:

Since, $a^2 - 25 = (a-5)(a+5) $
$ a^2 -2a -35 = a^2 -7a +5a -35 $
                         $= a(a-7)+5(a-7) $
                         $= (a+5)(a-7) $
and
$a^2+ 12a + 35 =a^2 +7a +5a +35 $
                          $=(a+7)(a+5) $
Clearly HCF of $a^2 - 25, a^2 -2a -35$ and $a^2+ 12a + 35$ i.e $ (a-5)(a+5), (a+5)(a-7)$ and $(a+7)(a+5)$ is $a+5$
Option D is correct.

H.C.F. of $x^3 -1$ and $x^4 + x^2 + 1$ is

maths real number real numbers on number line fundamental theorem of arithmetic common factors and hcf

  1. $x^2+x+1$

  2. $x-1$

  3. $x^3-1$

  4. $x^4+x^2+1$

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Correct Option: A
Explanation:

$x^3 -1=(x-1)(x^2+x+1)$
$x^4+x^2+1=(x^2+x+1)(x^2-x+1)$
Clearly H.C.F of $x^3 -1$ and $x^4 + x^2 + 1$
i.e.H.C.F of $(x-1)(x^2+x+1)$ and $(x^2+x+1)(x^2-x+1)$ is  $(x^2+x+1)$
Option A is correct.

H.C.F. of $x^2-1$ and $x^3-1$ is

maths real number real numbers on number line fundamental theorem of arithmetic common factors and hcf

  1. $(x^2-1)^2$

  2. $(x-1)$

  3. x+1

  4. $x^2+x+1$

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Correct Option: B
Explanation:

$x^2-1 = (x + 1) (x -1)$
$x^3 -1 = (x -1) (x^2 + x + 1)$
$\therefore H.C.F. = (x - 1)$
Option B is correct.

Find the HCF of $x^3y^2, x^2y^3$ and $x^4y^4$

maths real number real numbers on number line fundamental theorem of arithmetic common factors and hcf

  1. $x^3y^4$

  2. xy

  3. $x^2y^2$

  4. $x^4y^4$

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Correct Option: C
Explanation:

$x^3y^2=x^3\times y^2$
$x^2y^3=x^2\times y^3$
and $x^4y^4=x^4\times y^4$
$\therefore$ HCF $=x^2\times y^2=x^2y^2$.
Option C is correct.

The LCM of 54 90 and a third number is 1890 and their HCF is 18 The third number is

maths real number real numbers on number line fundamental theorem of arithmetic common factors and hcf

  1. 36

  2. 180

  3. 126

  4. 108

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Correct Option: C
Explanation:

Given the LCM two numbers 54, 90 and third number is 1890 and HCF is 18

Let the number is 18x because one factor is also 18 the common factor HCF
Then factor 54,90 ,18 =$18\times 3,18\times 5,18\times 18\times x$

$\therefore 18\times 3\times 5\times x=1890\Rightarrow 270x=1890\Rightarrow x=7$
Then third number is $18\times 7=126$