N= 2^4 x3x7^5 PN = K, where P is an integer and K is a square number.

Find the smallest value of P.

The smallest value of P is 21

Step-by-step explanation:

The given equations is presented as follows;

N = 2⁴ × 3 × 7⁵

P·N = K

The given variables are;

'P', and 'K'

Where;

P = An integer

K = x²

x = An integer

N = 2⁴ × 3 × 7⁵ = 806736

Therefore, we have;

P·N = K = P × 806736 = K = x²

P × 2⁴ × 3 × 7⁵ = P × 2⁴ × 3 × 7⁴ × 7 = K = x²

When P = 21 = 3 × 7, we have;

P × 2⁴ × 3 × 7⁴ × 7 = 3 × 7 × 2⁴ × 3 × 7⁴ × 7 = x²

Which gives;

3 × 3 × 7 × 7 ×  2⁴ × 7⁴ = x²

3² × 7² × 2⁴ × 7⁴ = x²

3² × 2⁴ × 7⁶ = x²

x = √(3² × 2⁴ × 7⁶) = 3 × 2² × 7³ = 4116

K = x² = 4116² = 16941456

P × N = 21 × 2⁴ × 3 × 7⁵ = 16941456

Therefore, the smallest value of P = 21.

8p+(-9)

(-9)3p+5p and you can swith em around