Answer:
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.
