A prime number is an integer greater than 1 that is evenly
divisible by only 1 and itself. For example, 2, 3, 5, and 7 are
prime numbers, but 4, 6, 8, and 9 are not. Create a PrimeNumber
application that prompts the user for a number and then displays a
message indicating whether the number is prime or not. Hint: The %
operator can be used to determine if one number is evenly divisible
by another. Java
A prime number is an integer greater than 1 that is evenly
divisible by only 1 and itself. For example, 2, 3, 5, and 7 are
prime numbers, but 4, 6, 8, and 9 are not. Create a PrimeNumber
application that prompts the user for a number and then displays a
message indicating whether the number is prime or not. Hint: The %
operator can be used to determine if one number is evenly divisible
by another.
b) Modify the...
a. Use mathematical induction to prove that for any positive
integer ?, 3 divide ?^3 + 2?
(leaving no remainder).
Hint: you may want to use the formula: (? + ?)^3= ?^3 + 3?^2 * b +
3??^2 + ?^3.
b. Use strong induction to prove that any positive integer ? (? ≥
2) can be written as a
product of primes.
A prime number is an integer greater than 1 that is evenly
divisible by only 1 and itself. For example, the number 5 is prime
because it can only be evenly divided by 1 and 5. The number 6,
however, is not prime because it can be divided by 1, 2, 3, and
6.Write a Boolean function named isPrime, which takes an integer
as an argument and returns true if the argument is a prime number,
and false otherwise. Demonstrate...
Use
mathematical induction to prove that for every integer n >=2, if
a set S has n elements, then the number of subsets of S with an
even number of elements equals the number of subsets of S with an
odd number of elements.
pleases send all detail solution.
Let p be an integer other than 0, ±1.
(a) Prove that p is prime if and only if it has the property
that whenever r and s are integers such that p = rs, then either r
= ±1 or s = ±1.
(b) Prove that p is prime if and only if it has the property
that whenever b and c are integers such that p | bc, then either p
| b or p | c.
12 pts) Use Mathematical Induction to prove that
an=n3+5n is divisible by 6 when ever
n≥0. You may explicitly use without proof the fact that the product
n(n+1) of consecutive integers n and n+1 is
always even, that is, you must state where you use this fact in
your proof.Write in complete sentences since this is an induction
proof and not just a calculation. Hint:Look up Pascal’s
triangle.
(a)
Verify the initial case n= 0.
(b)
State the induction hypothesis....