find the least number by which 10368 should be (1) increase (2) decreased (3) multiplied (4) divided to make it perfect square.

Question

i am not getting this question answwer

Answer 1

The prime factors of 10368 are:

  10368  =  2*2*2*2*2*2*2*3*3*3*3

 

  10368  =  ( (2*2*2)*(3*3) ) * ( (2*2*2*2)*(3*3) )

 

Rule to remember:  Perfect squares have an even number of each prime factor.

 

 

We can make a perfect square by multiplying bt 2 (the least prime factor that appears an odd number of times).

   20736 = ( (2* 2*2*2)*(3*3) ) * ( (2*2*2*2)*(3*3) )

 

Or, we can divide it by 2 (the least prime factor that appears an odd number of times).

   5184 = ( (2*2*2)*(3*3) ) * ( (    2*2*2)*(3*3) )

 

Since SQRT(10368) = 101.82  is between 101*101=10201  and 102*102=10404,  (the nearest perfect squares),

 

we may add 36  to get 10404,

 

  or subtract 167 to get 10101.

Answer 2

I solved these numerically using www.desmos.com and MS Excel

 

Addition:

 

y = sqrt(10368+x) must be a POSITIVE integer for x>0

The graph of this function contains the point (36,102);

 sqrt( 10368+36) = 102

 Add 36

 

 

 

Subtraction

  y = sqrt(10368-x) must be a positive integer for x>0;

The graph of this function contains (167,101);

 sqrt ( 10368 - 167) = 101 

Subtract 167

 

 

Multiply:  is the easiest..... ZERO

 

 

The division is the one that is not cut and dry.... there is no SMALLEST number to divide by...

 THere are INFINITELY many...

 

As x gets closer to zero, the square root of 10368/x grows infinitely large and blows up

For example at x=1/2, the square root is 144, a perfect square

At around x=0.0648 the square root is 400 another perfect square...

 

By examining the graph of y = sqrt( 10368/x) there are infinitely many perfect

squares that can be generated by picking x sufficiently close to zero.

 

The best we can do is derive a formula for calculating the x-coordinate needed to generate

the perfect square we want.

 

  For perfect square y:    y = sqrt(10368/x)  <--- must invert this function

                                    y^2 = 10368/x   <--- squares both sides

                                    x = 10368/y^2  <---- solves for x                                     

 

 So plug in the perfect square that you want into this formula for y, and it will tell you what x-coordinate

 required to get it.

 

 For example,  y=144,  x = 10368/144^2 = 1/2 as stated above

                     y =400, x = 0.0648 as stated above

                      y=10000, x = 0.00010368

                     

                                  

 

 

Answer 3

The best i can come up with is if you multiply by 2, you get 20736 which is 144 squared