Question

N= 3 4 x7x5 5 PN = K, where P is an integer and K is a square number. Find the smallest value of P.

Answer 1

Given: N= 3 4 x7x5 5 PN = K, where P is an integer and K is a square number

To find: The smallest value of P.

Solution:

The Given equation is represented as follows:

N=$3^{4}$ X 7 X $5^{5}$

P. N= K

The given variables are:

'P' & 'N'

K=x²

x=an integer

N=$3^{4}$ X 7 X $5^{5}$

  =81 X 7 X 3125

  =1,771,875

Therefore, PN=K=P X 1,771,875=K=x²

when P=35=7 X 5=,we have,

P X $3^{4}$ X 7 X $5^{5}$ X$5^{4}$ = 7 X 5 X  $3^{4}$ X 7 X 5 X $5^{4}$ =  x²

which gives

7 X 7 X 5 x 5 X $3^{4}$ X$5^{4}$ =  x²

${7^{2} X 5^{6} X 3^{4}= x^{2}$

x = √(7² X $5^{6}$ X $3^{4}$)

    = 7 X 5² X 3²

    =7875

K=x² =7875²

      =62,015,625

P x N= 35 X $3^{4}$ X 7 X $5^{5}$ =62,015,625

Therefore, the smallest value of P = 35