HCF, LCM and Common Factors - Class X

Factors of $ 9 = 1, 3, 9 $
Factors of $ 36 = 1, 3, 4, 6, 9, 12, 36 $

Common factors are $ 1, 3, 9 $

Common factor of 950 and 570 is 190 , So A is the Correct answer

Factors of 25 = 1, 5 and 25.
Factors of 75 = 1,3,5,15,25 and 75.
Factors of 100 = 1,2,4,5,10,20,25,50 and 100
Therefore, common factor of 25,75 and 100 = 1,5,25.
H.C.F of 25,75 and 100 = 25.
Option B is correct.

To  find  the  Highest  Common  Factor  (HCF)  of  two  or  more  numbers, 
find  prime  factors  of  the  numbers , and  then  identify  the  common  prime  factors.
Then  the  HCF  is  the  product  of  the  common  prime  factors.
$3^{5}= 1 \times 3^{5}$
$3^{9} = 1 \times 3^{5} \times 3^{4}$
$3^{14} = 1 \times 3^{5} \times 3^{9}$
Hence the HCF is $ 3^{5}.$
Another  method  to  find  the  answer  is  to find  the  largest  divisor  of all  three  numbers,
 which  is $ 3^{5} $

(A) Factors of 16 are $1, 2, 4, 8, 16$
(B) Factors of 14 are $1, 2, 7, 14$
(C) Factors of  18 are $1, 2, 3, 6, 9, 18$

$6 abc$, $24ab^2$, $12 a^2b$
The factors of $6abc=2\times 3\times a\times b\times c$
The factors of $24ab^2=2\times 2\times 2\times 3\times a\times b\times b$
The factors of $12 a^2b=2\times 2\times 3\times a\times a\times b$
The common factors are $2\times 3\times a\times b=6ab$

$2x$, $3x^2$, $4$
The factors of $2x=2\times x$
The factors of $3x^2=3\times x\times x$
The factors of $4=2\times 2$
Thus, the common factors is $1$

$2y$,$22xy$
The factors of $2y=2\times y$
The factors of $22xy=2\times 11\times x\times y$
Thus, the common factors are $2\times y=2y$

$16 x^3$,  $4x^2$, $32x$
The factors of $16 x^3=2\times 2\times 2\times 2\times x\times x\times x$
The factors of $4x^2=2\times 2 \times x\times x$
The factors of $32x=2\times 2\times 2\times 2\times 2\times x$
The common factors are $2\times 2 \times x=4x$

Factors of $ 18 = 1, 2, 3, 6, 8, 9$ and $ 18 $

Factors of $ 24 = 1, 2, 3, 4, 6, 8, 12 $ and $ 24 $

Common factors are $ 1, 2, 3, 6 $

Factors of $ 50 = 1, 2, 5, 10, 25 $ and $ 50 $

Factors of $ 75 = 1,3, 5, 15, 25 $ and $ 75 $

Common factors are $ 1, 5, 25 $

Factors of $ 9 $  are $ 1, 3, 9 $
Factors of $ 36 $ are $ 1, 3, 4, 6, 9, 12, 36 $
$\therefore $ common factors are $ 1, 3, 9 $.

L.C.M of $2352=2^2\times 2^2\times 7^2\times 3$
To make $2352$ a perfect square it must be multiplied by $3$

$1$st common multiple of $6,8$ is same as LCM of these numbers.

$12= 2^{2}\times 3$

Numbers $= 12, 15, 21$

 Number which is exactly divisible by 20, 28, 32 and 35 should be the common multiple of all these.
$20 = 2^2 \times 5$
$28 = 2^2 \times 7 \Rightarrow 32 = 2^5$
35.=5 $\times $ 7
LCM = $2^5 \times 5 \times 7 = 1120$
Hence the greatest number that should be subtracted
=5834-1120 = 4714