Tag: fundamental theorem of arithmetic

Questions Related to fundamental theorem of arithmetic

Mark the correct alternative of the following.
The HCF of $100$ and $101$ is _________.

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

  1. $1$

  2. $7$

  3. $37$

  4. None of these

Show answer
Correct Option: A
Explanation:

The given number $100$ and $101$ are consecutive numbers and respectively even and odd numbers.

So the HCF is $1$.

If HCF of $210$ and $55$ is of the form $(210) (5) + 55 y$, then the value of $y$ is :

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

  1. $-19$

  2. $-18$

  3. $5$

  4. $55$

Show answer
Correct Option: A
Explanation:

  HCF of 210 and 55 is 5

Given HCF = $(210)(5)+55y$
$5 =(210)(5)+55y$
On solving this equation we get $y=-19$
 So correct answer will be option A

When the HCF of $468$ and $222$ is written in the form of  $ 468 x + 222y$ then the value of $ x$ and $y$ is 

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

  1. $x =-9 \ and \ y =19$

  2. $x =9 \ and \ y = -19$

  3. $x =9\  and \ y = 19$

  4. $x =-9 \ and \ y =- 19$

Show answer
Correct Option: A
Explanation:

HCF of $468$ and $222$
$468 = \left(222 \times 2\right) + 24$
$222 = \left(24\times\ 9\right) + 6$
$24 = \left(6\times\ 4\right) + 0$
$\therefore HCF = 6$

$6 = 222 - \left(24\times\ 9\right)$
$ = 222 - \left[\left(468 -222 \times 2\right) \times\ 9\right]  $ [where $468 = 222 \times 2 + 24$]
$ = 222 - \left[468 \times 9 -222 \times 2 \times 9\right]$
$= 222 - \left(468 \times9\right) - \left(222\times 18\right)$
$ = 222 + \left(222 \times 18\right) - \left(468 \times9\right)$
$= 222\left[1 + 18\right]-  468 \times 9$
$= 222 \times19-  468 \times 9$
$  = 468 \times -9 + 222\times 19$
$\therefore x=-9$ and $y=19$.

If the H.C.F. of $A$ and $B$ is $24$ and that of $C$ and $D$ is $56,$ then the H.C.F. of $A, B, C$ and $D$ is

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

  1. $4$

  2. $12$

  3. $8$

  4. $3$

Show answer
Correct Option: C
Explanation:

Given the H.C.F. of $A$ and $B$ is $24$ and that of $C$ and $D$ is $56.$
Then the H.C.F. of $A, B, C$ and $D$ is the HCF of $24$ and $56$ which is $8.$

The HCF of $136 ,170 \ and \ 255$ is 

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

  1. $13$

  2. $15$

  3. $17$

  4. $1$

Show answer
Correct Option: C
Explanation:

136)170(1

  -    136
-------------------
          34)136(4
                136
----------------------------
                 0</div>

34)255(7
   -  238
-------------------
        17)34(2
             34
------------------------
              0

The H.C.F. of two expressions is x and their L.C.M is $ \displaystyle x^{3}-9x  $  IF one of the expression is $ \displaystyle x^{2}+3x  $  then,the other expression is 

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

  1. $ \displaystyle x^{2}-3x $

  2. $ \displaystyle x^{3}-3x $

  3. $ \displaystyle x^{2}+9x $

  4. $ \displaystyle x^{2}-9x $

Show answer
Correct Option: A
Explanation:

Let two expressions $p(x)$ and $q(x)$ then

$p(x)\times q(x)=L.C.M.\ \times\ H.C.F.$

Since $p(x)=x^2+3x$
$(x^2+3x)\times q(x)=(x^3-9x) \times\ x$
$(x^2+3x)\times q(x)=(x^2-3x) \times\ (x^2+3x)$

$q(x)=(x^2-3x)$
Hence, this is the required solution.

The HCF of the numbers $0.48, 0.72$ and $0.108$  is

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

  1. $1$

  2. $12$

  3. $0.12$

  4. $0.012$

Show answer
Correct Option: D
Explanation:

$480=2^5\times 3\times 5$
$720=2^4\times 3^2\times 5$
$108=2^2\times 3^2$
$HCF=2^2\times 3=12$

HCF of $0.48, 0.72$ and $0.108 = 0.012$
So, option $D$ is correct.

The the HCF of $248$ and $492$ is equal to

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

  1. $2$

  2. $3$

  3. $4$

  4. $5$

Show answer
Correct Option: C
Explanation:

$248=2^3 \times 31$

$492=2^2 \times 3 \times 41$
HCF is $4.$

Find the HCF of $26$ and $455$

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

  1. $11$

  2. $12$

  3. $13$

  4. none of the above

Show answer
Correct Option: C
Explanation:

$26=13\times 2$

$455=5\times 13\times 7$
HCF is $13.$

The H.C.F. of the numbers $16.5, 0.90$ and $15$ is

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

  1. $16.5$

  2. $0.90$

  3. $15$

  4. $0.3$

Show answer
Correct Option: D
Explanation:

$1650 = 2\times 3\times 5^{2}\times 11$
$90 = 2\times 3^{2} \times 5^{1}$
$1500 = 2^{2} \times 3^{1} \times 5^{3}$
H.C.F. of $1650, 90$ and $1500$ is $2\times 3\times 5 = 30$

Therefore, H.C.F. of $16.5, 0.90$ and $15$ is $0.30$.
So, option D is correct.