Homework SourseLet nab where b and a are positive integers prove a and b ca
Let n=ab where b and a are positive integers prove a and b can\'t be bigger than the root
Let n=ab where b and a are positive integers prove a and b can\'t be bigger than the root
Solution
To understand this, firstly I would establish some machinery: -
1) If m>0 and n>0 then mn>0
2) If m>n>0 and p>0 then (m-n)>0, p>0 ----> (m-n)p > 0. Hence mp > np
3) If m > n > 0 and p > x > 0 then mp > np > 0 and pn > xn > 0 hence mp > xn > 0
This establishes that inequalities can be multiplied together, as long as we pay attention to signs and symbols.
So, if we go by the DeMorgan\'s Law that \"If a > sqrt(n) and b > sqrt(n), then n != a*b\". This implies that ab > n, contradicting our hypothesis n = ab.
Hence it cannot be the case that both of a, b are strictly greater than sqrt(n). Logical negation of this proposition is that at least one of a, b are less than or equal to sqrt(n)
