Answer:
Lets say that P(n) is true if n is a prime or a product of prime numbers. We want to show that P(n) is true for all n > 1.
The base case is n=2. P(2) is true because 2 is prime.
Now lets use the inductive hypothesis. Lets take a number n > 2, and we will assume that P(k) is true for any integer k such that 1 < k < n. We want to show that P(n) is true. We may assume that n is not prime, otherwise, P(n) would be trivially true. Since n is not prime, there exist positive integers a,b greater than 1 such that a*b = n. Note that 1 < a < n and 1 < b < n, thus P(a) and P(b) are true. Therefore there exists primes p1, ...., pj and pj+1, ..., pl such that
p1*p2*...*pj = a
pj+1*pj+2*...*pl = b
As a result
n = a*b = (p1*......*pj)*(pj+1*....*pl) = p1*....*pj*....pl
Since we could write n as a product of primes, then P(n) is also true. For strong induction, we conclude than P(n) is true for all integers greater than 1.
It is "FALSE". In the problem, the expression is a^b where the variable "a" is called the base and the variable "b" is called the "EXPONENT or power". So, from the given problem, it is incorrect the "b" is called numerator.
Answer:
1: 50/4 = 12.5 | 40 x 12.5 = 500 | 160 x 12.5 = 2000 | 200 x 12.5 = 2500
2: 12.5
3: 750
4: 56 x 12.5 = 700, answer is 56
Area of triangle = base x height over 2
triangle 1
10 x 12 = 120
120/2 = 60
triangle 2
10 x 12 = 120
120/2 = 60
triangle 3
18 x 12 = 216
216/2 = 108
triangle 4
18 x 12 = 216
216/2 = 108
add
108 + 108 + 60 + 60 = 336 square units
2^12x-10
Explanation: (the picture)