Tag: hcf

Questions Related to hcf

The greatest common divisor of $878787878787$ and $787878787878$ equals.

maths hcf-lcm common factors and hcf hcf highest common factor (h.c.f.)

  1. $3$

  2. $9$

  3. $27$

  4. $101010101010$

  5. $303030303030$

Show answer
Correct Option: E
Explanation:

$787878787878)878787878787(1\ \quad \quad \quad \quad \quad  -\underline { 787878787878 } \ \quad \quad \quad \quad \quad \quad \quad 90909090909)787878787878(8\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad  \underline { -727272727272 } \ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad 60606060606)90909090909(1\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \underline { -60606060606 } \ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad 30303030303)60606060606(2\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \underline { -60606060606 } \ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad  \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad 0$

$\therefore$ G.D.C = 30303030303

Three bells, toll at intervals of $36$ sec, $40$ sec and $48$ sec respectively. They start ringing together at particular time. They next toll together after

maths hcf-lcm common factors and hcf hcf highest common factor (h.c.f.)

  1. $6$ minutes

  2. $12$ minutes

  3. $18$ minutes

  4. $24$ minutes

Show answer
Correct Option: B
Explanation:

G.C.D of $36,40,48=720\Rightarrow 720sec=12min$
$\therefore$ Next time when three balls toll together is after $12$ mins

The G.C.D. of two whole numbers is $5$ and their L.C.M. is $60$. If one of the numbers is $20$, then the other number would be

maths hcf-lcm common factors and hcf hcf highest common factor (h.c.f.)

  1. $23$

  2. $13$

  3. $16$

  4. $15$

Show answer
Correct Option: D
Explanation:

If we are given two numbers $N _1$ and $N _2$ and their $G.C.D$ and $L.C.M.$.

then by property of numbers $N _1$$\times$$N _2=G.C.D$ $\times$ $L.C.M.$

Here Given:
$N _1=20$
$G.C.D.=5$
and $L.C.M=60$
Let, $N _2=x$

then from  above relation
$20$$\times$$x=5$$\times$$60$

$=>x=\dfrac{300}{20}$

$=>x=15$

The HCF of $2{x^2}$ and $12{x^2}$ is

maths hcf-lcm common factors and hcf hcf highest common factor (h.c.f.)

  1. $2{x^2}$

  2. $12{x^2}$

  3. $2x$

  4. $12x$

Show answer
Correct Option: A
Explanation:

$2x^2=2\times x\times x$


$12x^2=6\times 2\times x\times x$

          $=3\times 2\times 2 \times x\times x$

Common factor between $2x^2$ and $12x^2=2\times x\times x=2x^2$

$\therefore$  H.C.F of $2x^2$ and $12x^2$ is $2x^2$.

Find the HCF of $25$ and $30$.

maths hcf-lcm common factors and hcf hcf highest common factor (h.c.f.)

  1. $25$

  2. $6$

  3. $1$

  4. $5$

Show answer
Correct Option: D
Explanation:

$\Rightarrow$$25=5\times5$

$\Rightarrow$$30=5\times6$

Hence the Highest common factor is 5

Therefore HCF is $5$

The solution of: $8\mod x\equiv 6\mod 14$ is,

maths hcf-lcm common factors and hcf hcf highest common factor (h.c.f.)

  1. ${8, 6}$

  2. ${6, 14}$

  3. ${6, 13}$

  4. ${8, 14}$

Show answer
Correct Option: C
Explanation:
Solution:-
$8x \equiv 6 \left( mod \ 14 \right)$

$\because \; gcd \left( 8, 14 \right) = 2 \text{ divides } 6$

To find solutions, we first solve

$8x − 14y = 6$

By trial and error method, we find a solution

$\left( x, y \right) = \left( 6, 3 \right)$

This means that $x \equiv 6 \left( mod \ 14 \right)$ is a solution

To the congruence $8x \equiv 6 \left( mod \ 14  \right)$

$\therefore$ Incongruent solutions are,

$x = 6 +\left ( k \times \dfrac{14}{2} \right );  k = 0, 1$

$\therefore \; x = 6, 13$

Hence option $C$ is the answer.

If $G.C.D\ (a , b) = 1$ then $G.C.D\ ( a+b , a-b )$=?

maths hcf-lcm common factors and hcf hcf highest common factor (h.c.f.)

  1. $1$ or $2$

  2. $a$ or $b$

  3. $a+b$ or $a-b$

  4. $4$

Show answer
Correct Option: A
Explanation:
It is given that GCD$\left(a,b\right)=1$

Let GCD$\left(a-b,a+b\right)=d$

$\Rightarrow\,d$ divides $a-b$ and $a+b$

there exists integers $m$ and $n$ such that 

$a+b=m\times d$        ..........$(1)$

and $a-b=n\times d$        ..........$(2)$

Upon adding and subtracting equation $(1)$ and $(2)$ we get

$2a=\left(m+n\right)\times d$         ..........$(3)$

and $2b=\left(m-n\right)\times d$         ..........$(4)$

Since, GCD$\left(a,b\right)=1$(given)

$\therefore\,2\times GCD\left(a,b\right)=2$

$\therefore\,GCD\left(2a,2b\right)=2$ since $GCD\left(ka,kb\right)=kGCD\left(a,b\right)$

Upon substituting  value of $2a$ and $2b$ from equations $(3)$ and $(4)$ we get

$\therefore\,gcd\left(\left(m+n\right)\times d,\left(m-n\right)\times d\right)=2$

$\therefore\,d\times gcd\left(\left(m+n\right),\left(m-n\right)\right)=2$

$\therefore\,d\times$ some integer$=2$

$\therefore\,d$ divides $2$

$\therefore\,d\le 2$ if $x$ divides $y,$ then $\left|x\right|\le \left|y\right|$

$\therefore\,d=1$ or $2$ since, gcd is always a positive integer.

ILLUSTRATION  2 The total number of factors (exculding 1) of 2160 is 

maths hcf-lcm common factors and hcf hcf highest common factor (h.c.f.)

  1. 40

  2. 39

  3. 41

  4. 38

Show answer
Correct Option: A

The GCD of two numbers is $17$ and their LCM is $765$. How many pairs of values can the numbers assume?

maths hcf-lcm common factors and hcf hcf highest common factor (h.c.f.)

  1. $1$

  2. $2$

  3. $3$

  4. $4$

Show answer
Correct Option: B
Explanation:

Since the GOD of numbers is $17$. So, the numbers are $17a$ and $17b$, where a and b are relatively prime.
LCM$=765$

$\Rightarrow 17a\times 17b=765$
$\Rightarrow ab=45$
$\Rightarrow a=15, b=9$ or $a=9$, $b=5$.
So, the numbers are $17\times 5=85$ and $17\times 9=153$.
The numbers can be $17\times 1=17$ and $765$. So, two pairs are possible.

If two positive integers $a$ and $b$ are written as $a=x^3y^2$ and $b=xy^3$; $x, y$ are prime numbers, then HCF of $a$ and $b$ is

maths hcf-lcm common factors and hcf hcf highest common factor (h.c.f.)

  1. $xy$

  2. $xy^2$

  3. $x^3y^3$

  4. $x^2y^2$

Show answer
Correct Option: B
Explanation:

Given,

$a={  x}^{3  }{ y }^{2  } = x\times x\times x\times y\times y$

$b={  x}{ y }^{3  }         =x\times y\times y\times y$

H.C.F of $a,b$  = ${  x}{ y }^{2  } $